Recent zbMATH articles in MSC 12K10, 14T, 52https://zbmath.org/atom/cc/52,14T,12K102023-03-23T18:28:47.107421ZWerkzeugInfinite jigsaws, upside-down fractals and irrational slices: finding order without periodicityhttps://zbmath.org/1503.000232023-03-23T18:28:47.107421Z"Walton, James J."https://zbmath.org/authors/?q=ai:walton.james-jSummary: Periodic decorations have been familiar to both mathematicians and artists for millennia. However, periodicity is not necessary for long-range order, a fact whose associated mathematical theory was only initiated more
recently. This article explores what aperiodic order is, how to construct examples of aperiodic order and how mathematicians can begin studying it.Abstract 3-rigidity and bivariate \(C_2^1\)-splines. II: Combinatorial characterizationhttps://zbmath.org/1503.050172023-03-23T18:28:47.107421Z"Clinch, Katie"https://zbmath.org/authors/?q=ai:clinch.katie"Jackson, Bill"https://zbmath.org/authors/?q=ai:jackson.bill"Tanigawa, Shin-ichi"https://zbmath.org/authors/?q=ai:tanigawa.shin-ichiSummary: We showed in the first paper of this series that the generic \(C_2^1\)-cofactor matroid is the unique maximal abstract \(3\)-rigidity matroid. In this paper we obtain a combinatorial characterization of independence in this matroid. This solves the cofactor counterpart of the combinatorial characterization problem for the rigidity of generic 3-dimensional bar-joint frameworks. We use our characterization to verify that the counterparts of conjectures of Dress (on the rank function, see [\textit{A. Dress} et al., ``Classification of mobile molecules by category theory'', in: Symmetries and properties of non-rigid molecules: a comprehensive survey. Amsterdam: Elsevier Science Ltd. 39--58 (1983)]) and \textit{L. Lovász} and \textit{Y. Yemini} [SIAM J. Algebraic Discrete Methods 3, 91--98 (1982; Zbl 0497.05025)] (which suggested a sufficient connectivity condition for rigidity) hold for this matroid.
For Part I see [the authors, ibid. 2022, Paper No. 2, 50 p. (2022; Zbl 07643619)].Small cocircuits in minimally vertically 4-connected matroidshttps://zbmath.org/1503.050192023-03-23T18:28:47.107421Z"Oxley, James"https://zbmath.org/authors/?q=ai:oxley.james-g"Walsh, Zach"https://zbmath.org/authors/?q=ai:walsh.zach-wSummary: \textit{R. Halin} [Math. Ann. 182, 175--188 (1969; Zbl 0172.25804)] proved that every minimally \(k\)-connected graph has a vertex of degree \(k\). More generally, does every minimally vertically \(k\)-connected matroid have a \(k\)-element cocircuit? Results of \textit{U. S. R. Murty} [Discrete Math. 8, 49--58 (1974; Zbl 0278.05027)] and \textit{Pak-Kan Wong} [J. Reine Angew. Math. 299/300, 1--6 (1978; Zbl 0367.05025)] give an affirmative answer when \(k\leq 3\). We show that every minimally vertically \(4\)-connected matroid with at least six elements has a \(4\)-element cocircuit, or a \(5\)-element cocircuit that contains a triangle, with the exception of a specific nonbinary \(9\)-element matroid. Consequently, every minimally vertically \(4\)-connected binary matroid with at least six elements has a \(4\)-element cocircuit.Flexing infinite frameworks with applications to braced Penrose tilingshttps://zbmath.org/1503.050262023-03-23T18:28:47.107421Z"Dewar, Sean"https://zbmath.org/authors/?q=ai:dewar.sean"Legerský, Jan"https://zbmath.org/authors/?q=ai:legersky.janSummary: A planar framework -- a graph together with a map of its vertices to the plane -- is flexible if it allows a continuous deformation preserving the distances between adjacent vertices. Extending a recent previous result, we prove that a connected graph with a countable vertex set can be realized as a flexible framework if and only if it has a so-called NAC-coloring. The tools developed to prove this result are then applied to frameworks where every 4-cycle is a parallelogram, and countably infinite graphs with \(n\)-fold rotational symmetry. With this, we determine a simple combinatorial characterization that determines whether the 1-skeleton of a Penrose rhombus tiling with a given set of braced rhombi will have a flexible motion, and also whether the motion will preserve 5-fold rotational symmetry.Diameter estimates for graph associahedrahttps://zbmath.org/1503.050292023-03-23T18:28:47.107421Z"Cardinal, Jean"https://zbmath.org/authors/?q=ai:cardinal.jean"Pournin, Lionel"https://zbmath.org/authors/?q=ai:pournin.lionel"Valencia-Pabon, Mario"https://zbmath.org/authors/?q=ai:valencia-pabon.mario-eSummary: Graph associahedra are generalized permutohedra arising as special cases of nestohedra and hypergraphic polytopes. The graph associahedron of a graph \(G\) encodes the combinatorics of the search trees on \(G\), defined recursively by a root \(r\) together with search trees on each of the connected components of \(G-r\). In particular, the 1-skeleton of the corresponding graph associahedron is the rotation graph of those search trees. We investigate the diameter of graph associahedra as a function of some graph parameters. We give a tight bound of \(\Theta (m)\) on the diameter of trivially perfect graph associahedra on \(m\) edges. We consider the maximum diameter of associahedra of graphs on \(n\) vertices and of given tree-depth, treewidth, or pathwidth, and give lower and upper bounds as a function of these parameters. We also prove that the maximum diameter of associahedra of graphs of pathwidth two is \(\Theta (n\log n)\). Finally, we give the exact diameter of the associahedra of complete split graphs and of unbalanced complete bipartite graphs.Every nonsingular spherical Euclidean distance matrix is a resistance distance matrixhttps://zbmath.org/1503.050302023-03-23T18:28:47.107421Z"Estrada, Ernesto"https://zbmath.org/authors/?q=ai:estrada.ernestoSummary: The communicability distance [\textit{E. Estrada}, ibid. 436, No. 11, 4317--4328 (2012; Zbl 1243.05071)] is a useful metric to characterize alternative navigational routes in graphs. Here we prove that it is the resistance distance between a pair of nodes in a weighted graph. We extend this result and prove that every nonsingular Euclidean distance matrix is the resistance distance matrix of a weighted graph. We briefly analyze some mathematical properties of the communicability Laplacian matrix which emerges from the current analysis.On the classification of motions of paradoxically movable graphshttps://zbmath.org/1503.050862023-03-23T18:28:47.107421Z"Grasegger, Georg"https://zbmath.org/authors/?q=ai:grasegger.georg"Legerský, Jan"https://zbmath.org/authors/?q=ai:legersky.jan"Schicho, Josef"https://zbmath.org/authors/?q=ai:schicho.josefA recent result [\textit{G. Grasegger} et al., Discrete Comput. Geom. 62, No. 2, 461--480 (2019; Zbl 1417.05178)] provides a combinatorial characterization of graphs that admit flexible planar realizations. The necessary and sufficient condition is the existence of an NAC coloring -- an assignment of two colors to the graph's edges such that every cycle is either monochromatic or has at least two edges of each color. The proof is constructive and provides a flexible graph embedding for a given NAC coloring. However, a proper realization, i.e., a realization where no two vertices coincide, cannot be guaranteed.
This article presents an algorithm to compute all proper flexible realizations of generically rigid graphs. The basic idea is to relate NAC colorings of four cycles to their respective motions (general, parallelogram, or anti-parallelogram motion, deltoid motion, or degenerate cases) and to ensure global consistency within the complete graph.
As testified by the authors' classification of motions of the graph \(Q_1\), the problem itself is quite complex. Currently, the algorithm is only feasible for reasonably simple graphs. It is, however, much more efficient than the preceding methods. This is demonstrated by the authors' light-handed derivation of flexible embeddings of the graph \(K_{3,3}\) -- a task that required heavy use of computer algebra in the original publication [\textit{D. Walter} and \textit{M. L. Husty}, ``On a nine-bar linkage, its possible configurations and conditions for paradoxical mobility'', in: Proceedings of the 12th world congress in mechanism and machine science, IFToMM '07. Fredericton, Canada: International Federation for the Promotion of Mechanism and Machine Science. 6 p. (2007)].
Reviewer: Hans-Peter Schröcker (Innsbruck)Reachability in arborescence packingshttps://zbmath.org/1503.050962023-03-23T18:28:47.107421Z"Hörsch, Florian"https://zbmath.org/authors/?q=ai:horsch.florian"Szigeti, Zoltán"https://zbmath.org/authors/?q=ai:szigeti.zoltanThe authors consider the packing of arborescences in this paper. First, they show how to translate the inductive concepts used in the latter two articles into a simple proof technique that allows for associating previous results on arborescence packings. Second, they verify that a strong version of Edmunds' theorem on packing spanning arborescences implies \textit{N. Kamiyama} et al.'s result on packing reachability arborescences [Combinatorica 29, No. 2, 197--214 (2009; Zbl 1212.05209)] and that \textit{O. D. de Gevigney} et al.'s theorem on matroid-based packing of arborescences [SIAM J. Discrete Math. 27, No. 1, 567--574 (2013; Zbl 1268.05165)] implies \textit{C. Király}'s result on matroid-reachability-based packing of arborescences [SIAM J. Discrete Math. 30, No. 4, 2107--2114 (2016; Zbl 1350.05138)]. Third, they infer a new result about matroid-reachability-based packing of mixed hyperarborescences from a theorem on matroid-based packing of mixed hyperarborescences due to \textit{Q. Fortier} et al. [Discrete Appl. Math. 242, 26--33 (2018; Zbl 1384.05119)]. Finally, they discuss the algorithmic aspects of the problems considered, they derive algorithms to find the desired packings of arborescences in all settings and utilize Edmonds' weighted matroid intersection algorithm to find solutions minimizing a given weight function.
Reviewer: Sizhong Zhou (Zhenjiang)Backtracking algorithms for constructing the Hamiltonian decomposition of a 4-regular multigraphhttps://zbmath.org/1503.051192023-03-23T18:28:47.107421Z"Korostil, A. V."https://zbmath.org/authors/?q=ai:korostil.alexander-v"Nikolaev, A. V."https://zbmath.org/authors/?q=ai:nikolaev.andrei-valerevich(no abstract)A separation between tropical matrix rankshttps://zbmath.org/1503.150372023-03-23T18:28:47.107421Z"Shitov, Yaroslav"https://zbmath.org/authors/?q=ai:shitov.yaroslav-nikolaevichSummary: We continue to study the rank functions of tropical matrices. In this paper, we explain how to reduce the computation of ranks for matrices over the `supertropical semifield' to the standard tropical case. Using a counting approach, we prove the existence of a \(01\)-matrix with many ones and without large all-one submatrices, and we put our results together and construct an \(n\times n\) matrix with tropical rank \(o(n^{0.5+\varepsilon})\) and Kapranov rank \(n-o(n)\).Some generator functions for \(s\)-convex functions in the fourth sensehttps://zbmath.org/1503.260182023-03-23T18:28:47.107421Z"Kemali, Serap"https://zbmath.org/authors/?q=ai:kemali.serapIn this paper, the author studies some generator functions for \(s\)-convex functions and their properties in the fourth sense, which are expressed via single integral or double integral representations. Examples are given to illustrate the results obtained to derive some special mean relations and the inequalities involving beta and digamma functions are presented and discussed.
Reviewer: James Adedayo Oguntuase (Abeokuta)Loomis-Whitney inequalities in Heisenberg groupshttps://zbmath.org/1503.280052023-03-23T18:28:47.107421Z"Fässler, Katrin"https://zbmath.org/authors/?q=ai:fassler.katrin-s"Pinamonti, Andrea"https://zbmath.org/authors/?q=ai:pinamonti.andreaSummary: This note concerns \textit{Loomis-Whitney inequalities} in Heisenberg groups \(\mathbb{H}^n\):
\[
|K| \lesssim \prod_{j=1}^{2n}|\pi_j(K)|^{\frac{n+1}{n(2n+1)}}, \quad K \subset \mathbb{H}^n.
\]
Here \(\pi_j\), \(j=1,\ldots ,2n\), are the \textit{vertical Heisenberg projections} to the hyperplanes \(\{x_j=0\} \), respectively, and \(|\cdot |\) refers to a natural Haar measure on either \(\mathbb{H}^n\), or one of the hyperplanes. The Loomis-Whitney inequality in the first Heisenberg group \(\mathbb{H}^1\) is a direct consequence of known \(L^p\) improving properties of the standard Radon transform in \(\mathbb{R}^2\). In this note, we show how the Loomis-Whitney inequalities in higher dimensional Heisenberg groups can be deduced by an elementary inductive argument from the inequality in \(\mathbb{H}^1\). The same approach, combined with multilinear interpolation, also yields the following strong type bound:
\[
\int_{\mathbb{H}^n} \prod_{j=1}^{2n} f_j(\pi_j(p))\,dp\lesssim \prod_{j=1}^{2n} \Vert f_j\Vert_{\frac{n(2n+1)}{n+1}}
\]
for all nonnegative measurable functions \(f_1,\ldots ,f_{2n}\) on \(\mathbb{R}^{2n} \). These inequalities and their geometric corollaries are thus ultimately based on planar geometry. Among the applications of Loomis-Whitney inequalities in \(\mathbb{H}^n\), we mention the following sharper version of the classical geometric Sobolev inequality in \(\mathbb{H}^n\):
\[
\Vert u\Vert_{\frac{2n+2}{2n+1}} \lesssim \prod_{j=1}^{2n}\Vert X_ju\Vert^{\frac{1}{2n}}, \quad u \in BV(\mathbb{H}^n),
\]
where \(X_j\), \(j=1,\ldots ,2n\), are the standard horizontal vector fields in \(\mathbb{H}^n\). Finally, we also establish an extension of the Loomis-Whitney inequality in \(\mathbb{H}^n\), where the Heisenberg vertical coordinate projections \(\pi_1,\ldots ,\pi_{2n}\) are replaced by more general families of mappings that allow us to apply the same inductive approach based on the \(L^{3/2}-L^3\) boundedness of an operator in the plane.Two problems on homogenization in geometryhttps://zbmath.org/1503.300592023-03-23T18:28:47.107421Z"Ivrii, Oleg"https://zbmath.org/authors/?q=ai:ivrii.oleg-v"Marković, Vladimir"https://zbmath.org/authors/?q=ai:markovic.vladimir|markovic.vladimir-mFor the entire collection see [Zbl 1478.00019].Some logarithmic Minkowski inequalities for nonsymmetric convex bodies and related problemshttps://zbmath.org/1503.310142023-03-23T18:28:47.107421Z"Ji, Lewen"https://zbmath.org/authors/?q=ai:ji.lewenSummary: In this paper, we show the existence of a solution to an even logarithmic Minkowski problem for \(p\)-capacity and prove some analogue inequalities of the logarithmic Minkowski inequality for general nonsymmetric convex bodies involving \(p\)-capacity.On arithmetic progressions in non-periodic self-affine tilingshttps://zbmath.org/1503.370362023-03-23T18:28:47.107421Z"Nagai, Yasushi"https://zbmath.org/authors/?q=ai:nagai.yasushi"Akiyama, Shigeki"https://zbmath.org/authors/?q=ai:akiyama.shigeki"Lee, Jeong-Yup"https://zbmath.org/authors/?q=ai:lee.jeong-yupSummary: We study the repetition of patches in self-affine tilings in \(\mathbb{R}^d\). In particular, we study the existence and non-existence of arithmetic progressions. We first show that an arithmetic condition of the expansion map for a self-affine tiling implies the non-existence of certain one-dimensional arithmetic progressions. Next, we show that the existence of full-rank infinite arithmetic progressions, pure discrete dynamical spectrum, and limit-periodicity are all equivalent for a certain class of self-affine tilings. We finish by giving a complete picture for the existence or non-existence of full-rank infinite arithmetic progressions in the self-similar tilings in \(\mathbb{R}^d\) .Orthogonality types in normed linear spaceshttps://zbmath.org/1503.460092023-03-23T18:28:47.107421Z"Alonso, Javier"https://zbmath.org/authors/?q=ai:alonso.javier"Martini, Horst"https://zbmath.org/authors/?q=ai:martini.horst"Wu, Senlin"https://zbmath.org/authors/?q=ai:wu.senlinThis chapter makes part of the first volume of ``Surveys in Geometry'', which presents research topics in geometry in a broad sense. The main theme of this chapter is the concept of orthogonality in normed linear spaces with particular attention to the geometric side of the subject. An extensive list of references is given, providing a good overview of the relevant material.
The chapter starts with a summary of definitions for orthogonality in normed linear spaces, all of them generalizing in a specific sense the well-known concept of orthogonality in real inner product spaces (Section 4.2). In the subsequent sections (4.3 up to 4.8), the most important properties of the orthogonality relation are investigated in detail for Roberts orthogonality, Birkhoff orthogonality, James and isosceles orthogonality and Pythagorean orthogonality. Much attention is paid to theorems concerning additional properties of the orthogonality (symmetry, homogeneity, additivity, etc.) forcing the normed space to be an inner product space. In Section 4.9 further types of orthogonality are being discussed and various open problems are posed.
The final Section 4.10 gives a survey of more recent results scattered in the literature but brought together here in a clear way.
For the entire collection see [Zbl 1481.51004].
Reviewer: Dirk Keppens (Gent)A groupoid approach to interacting fermionshttps://zbmath.org/1503.460412023-03-23T18:28:47.107421Z"Mesland, Bram"https://zbmath.org/authors/?q=ai:mesland.bram"Prodan, Emil"https://zbmath.org/authors/?q=ai:prodan.emilSummary: We consider the algebra \({{\dot{\Sigma }}}({{\mathcal{L}}})\) generated by the inner-limit derivations over the \({\text{GICAR}}\) algebra of a fermion gas populating an aperiodic Delone set \({{\mathcal{L}}}\). Under standard physical assumptions such as finite interaction range, Galilean invariance of the theories and continuity with respect to the deformations of the aperiodic lattices, we demonstrate that the image of \({{\dot{\Sigma }}}({{\mathcal{L}}})\) through the Fock representation can be completed to a groupoid-solvable pro-\(C^*\)-algebra. Our result is the first step towards unlocking the \(K\)-theoretic tools available for separable \(C^*\)-algebras for applications in the context of interacting fermions.On the Rockafellar function associated to a non-cyclically monotone mappinghttps://zbmath.org/1503.470692023-03-23T18:28:47.107421Z"Precupanu, Teodor"https://zbmath.org/authors/?q=ai:precupanu.teodorSummary: In an earlier paper [\textit{T. Precupanu}, J. Convex Anal. 24, No. 1, 319--331 (2017; Zbl 06704311)], we have given a definition of the Rockafellar integration function associated to a cyclically monotone mapping considering only systems of distinct elements in its domain. Thus, this function can be proper for certain non-cyclically monotone mappings. In this paper, we establish general properties of Rockafellar function if the graph of mapping does not contain finite set of accumulation elements where the mapping is not cyclically monotone. Also, some dual properties are given.Curvature measures and soap bubbles beyond convexityhttps://zbmath.org/1503.490342023-03-23T18:28:47.107421Z"Hug, Daniel"https://zbmath.org/authors/?q=ai:hug.daniel"Santilli, Mario"https://zbmath.org/authors/?q=ai:santilli.marioSummary: Extending the celebrated results of \textit{A. D. Alexandrov} (1958) [Zbl 0122.39601; Zbl 0269.53024; Zbl 0269.53023; Zbl 0119.16603] and \textit{A. Ros} [J. Differ. Geom. 27, No. 2, 215--220 (1988; Zbl 0638.53051)] for smooth sets, as well as the results of \textit{R. Schneider} [Comment. Math. Helv. 54, 42--60 (1979; Zbl 0392.52004)] and \textit{D. Hug} [Measures, curvatures and currents in convex geometry. Freiburg: Albert-Ludwigs-Universität (Habil.) (1999)] for arbitrary convex bodies, we obtain for the first time the characterization of the isoperimetric sets of a uniformly convex smooth finite-dimensional normed space (i.e. Wulff shapes) in the non-smooth and non-convex setting, based on a natural geometric condition involving the curvature measures. More specifically we show, under a weak mean convexity assumption, that finite unions of disjoint Wulff shapes are the only sets of positive reach \(A \subseteq \mathbb{R}^{n + 1}\) with finite and positive volume such that, for some \(k \in \{0, \ldots, n - 1\}\), the \(k\)-th generalized curvature measure \(\Theta_k^\phi(A, \cdot)\), which is defined on the unit normal bundle of \(A\) with respect to the relative geometry induced by \(\varphi\), is proportional to the top order generalized curvature measure \(\Theta_n^\phi(A, \cdot)\). If \(k = n - 1\) the conclusion holds for all sets of positive reach with finite and positive volume. We also prove a related sharp result about the removability of the singularities. This result is based on the extension of the notion of a normal boundary point, originally introduced by \textit{H. Busemann} and \textit{W. Feller} [Acta Math. 66, 1--47 (1936; JFM 62.0832.02)] for arbitrary convex bodies, to sets of positive reach. These findings are new even in the Euclidean space.
Several auxiliary and related results are proved, which are of independent interest. They include the extension of the classical Steiner-Weyl tube formula to arbitrary closed sets in a finite dimensional uniformly convex normed vector space, a general formula for the derivative of the localized volume function, which extends and complements recent results of \textit{A. Chambolle} et al. [Math. Z. 299, No. 3--4, 1257--1274 (2021; Zbl 1483.28001)], and general versions of the Heintze-Karcher inequality.The tiling book. An introduction to the mathematical theory of tilingshttps://zbmath.org/1503.520012023-03-23T18:28:47.107421Z"Adams, Colin"https://zbmath.org/authors/?q=ai:adams.colin-cThe book is devoted to the theory of tilings. Mostly tilings of the Euclidean plane are considered. The author essentially uses the classical monograph [\textit{B. Grünbaum} and \textit{G. C. Shephard}, Tilings and patterns. New York: W. H. Freeman and Company (1987; Zbl 0601.05001)]. The emphasis is placed on the interconnection of different disciplines of mathematics. In particular, isometries of the Euclidean plane as well as symmetry groups of tiles and tilings are discussed. The book contains a lot of nice color illustrations. \par The author used the material of the book for teaching a course in tiling theory. There are exercises and projects in the book. The appendix provides methods for creating interesting tilings.
Reviewer: Elizaveta Zamorzaeva (Chişinău)Isoperimetric inequalities in unbounded convex bodieshttps://zbmath.org/1503.520022023-03-23T18:28:47.107421Z"Leonardi, Gian Paolo"https://zbmath.org/authors/?q=ai:leonardi.gian-paolo"Ritoré, Manuel"https://zbmath.org/authors/?q=ai:ritore.manuel"Vernadakis, Efstratios"https://zbmath.org/authors/?q=ai:vernadakis.efstratiosIn this book, the authors consider the \textit{relative isoperimetric problem} in \textit{unbounded convex bodies} \(C\) (i.e., unbounded closed convex sets with non-empty interior in Euclidean space of dimension \(n \geqslant 2\)). No further assumptions are made on the regularity of \(\partial C\). The relative isoperimetric problem on \(C\) looks for sets \(E \subset C\) of given finite volume \(|E|\) minimizing the relative perimeter \(P_C(E)\) of \(E\) in the interior of \(C\). In this general context,
\[
P_C(E) := \sup \left\{\int_E \text{div} \xi \, d \mathcal H^n \colon \xi \in \Gamma_0(C), |\xi| \le 1\right\},
\]
where \(\Gamma_0(C)\) is the set of smooth vector fields with compact support in the interior of \(C\) and integration is done with respect to the \(n\)-dimensional Hausdorff measure. If both \(\partial C\) and \(\partial E\) are smooth, then \(P_C(E)\) is the standard \((n-1)\)-dimensional Hausdorff measure of \(\partial E\) in the interior of \(C\).
The \textit{isoperimetric profile} of \(C\) is defined by
\[
I_C(v) := \inf \left\{P_C(E) \colon E \subset C, |E| = v\right\}.
\]
The function \(I_C(v)\) provides an optimal isoperimetric inequality on \(C\) since \(P_C(F) \geqslant I_C(|F|)\) for any set \(F \subset C\). The sets for which this inequality turns into equality are called \textit{isoperimetric regions}. In this book, the relative isoperimetric problem is addressed by studying the properties of the isoperimetric profile.
The key content of the book is the following:
\begin{itemize}
\item Chapters 1 and 2 contain a literature overview and some preliminary results of general nature.
\item Chapter 3 introduces a class of unbounded convex bodies of \textit{uniform geometry}. More precisely, an unbounded convex body \(C\) is of uniform geometry if the volume of a relative ball \(B_C(x,r) = B(x,r) \cap C\) of a fixed radius \(r > 0\) cannot be made arbitrary small by letting \(x \in C\) go away to infinity. The rest of the chapter is devoted to studying the properties of such convex bodies.
\item In Chapter 4, a general result on existence of isoperimetric regions is proven. Namely, in Theorem 4.6, the authors show that the isoperimetric profile \(I_C(v)\) of an unbounded convex body \(C\) of uniform geometry is attained for any fixed \(v > 0\) by a \textit{generalized isoperimetric region} consisting of an array of sets \((E_0, \ldots, E_\ell)\) such that \(E_0 \subset C = K_0\) and \(E_i \subset K_i\) for suitable asymptotic cylinders \(K_1, \ldots, K_\ell\) which satisfy
\[
\sum_{i=0}^\ell |E_i| = v \quad \text{ and }\quad \sum_{i=0}^\ell P_{K_i}(E_i) = I_C(v).
\]
(Here, an \textit{asymptotic cylinder} is a local limit in the Hausdorff distance of a sequence of translations \(\{-x_j + C\}\), with \(\{x_j\}\subset C\) being a divergent sequence of points. It turns out that the uniform geometry condition is equivalent to the fact that any asymptotic cylinder is a convex body.)
\item In Chapter 5, the authors show that \(v \mapsto I_C(v)\) is a continuous function (Theorem 5.1), and provided \(C\) is of uniform geometry, \(v \mapsto I_C^{n/(n-1)}(v)\) is a concave function (Theorem 5.8).
\item Chapter 6 contains several new isoperimetric inequalities and rigidity results for the equality cases. For example, Theorem 6.3 establishes that for a convex body with non-degenerate asymptotic cone \(C_\infty\), the inequality \(I_C(v) \ge I_{C_\infty}(v)\) holds and the quotient \(I_C(v) / I_{C_\infty}(v) \to 1\) as \(v \to \infty\). The equality case is also analyzed (using re-scalings of large volumes in the cone). This result is used to show that \(I_C(v) / I_{C_{\min}}(v) \to 1\) as \(v \to 0\), where \(C_{\min}\) is a tangent cone to \(C\) or to an asymptotic cylinder of \(C\) with minimum solid angle (Theorem~6.9). Several corollaries of the last result are deduced later in the chapter.
\item The final chapter (Chapter 7) is dedicated to estimating the \textit{isoperimetric dimension} of \(C\). This quantity is defined as the number \(\alpha > 0\) such that there exist \(0 < \lambda_1 < \lambda_2\) so that
\[
\lambda_1 v \leqslant I_C^{\alpha/(\alpha - 1)}(v) \leqslant \lambda_2 v \quad \text{for all sufficiently large }v.
\]
In Theorem 7.4, the isoperimetric dimension for unbounded convex bodies \(C\) of uniform geometry is bounded in terms of the function \(b(r) = \inf_{x \in C}|B_C(x,r)|\).
\end{itemize}
Reviewer: Kostiantyn Drach (Klosterneuburg)Classifying simplicial dissections of convex polyhedra with symmetryhttps://zbmath.org/1503.520032023-03-23T18:28:47.107421Z"Betten, Anton"https://zbmath.org/authors/?q=ai:betten.anton"Mukthineni, Tarun"https://zbmath.org/authors/?q=ai:mukthineni.tarunSummary: A convex polyhedron is the convex hull of a finite set of points in \(\mathbb{R}^3\). A triangulation of a convex polyhedron is a decomposition into a finite number of 3-simplices such that any two intersect in a common face or are disjoint. A simplicial dissection is a decomposition into a finite number of 3-simplices such that no two share an interior point. We present an algorithm to classify the simplicial dissections of a regular polyhedron under the symmetry group of the prolyhedron.
For the entire collection see [Zbl 1496.68012].Algebraic polytopes in Normalizhttps://zbmath.org/1503.520042023-03-23T18:28:47.107421Z"Bruns, Winfried"https://zbmath.org/authors/?q=ai:bruns.winfriedSummary: We describe the implementation of algebraic polyhedra in Normaliz. In addition to convex hull computation/vertex enumeration, Normaliz computes triangulations, volumes, lattice points, face lattices and automorphism groups. The arithmetic is based on the package \textit{e-antic} by V. Delecroix.
For the entire collection see [Zbl 1496.68012].Practical volume estimation of zonotopes by a new annealing schedule for cooling convex bodieshttps://zbmath.org/1503.520052023-03-23T18:28:47.107421Z"Chalkis, Apostolos"https://zbmath.org/authors/?q=ai:chalkis.apostolos"Emiris, Ioannis Z."https://zbmath.org/authors/?q=ai:emiris.ioannis-z"Fisikopoulos, Vissarion"https://zbmath.org/authors/?q=ai:fisikopoulos.vissarionSummary: We study the problem of estimating the volume of convex polytopes, focusing on zonotopes. Although a lot of effort is devoted to practical algorithms for polytopes given as an intersection of halfspaces, there is no such method for zonotopes. Our algorithm is based on Multiphase Monte Carlo (MMC) methods, and our main contributions include: (i) a new uniform sampler employing Billiard Walk for the first time in volume computation, (ii) a new simulated annealing generalizing existing MMC by making use of adaptive convex bodies which fit to the input, thus drastically reducing the number of phases. Extensive experiments on zonotopes show our algorithm requires sub-linear number of oracle calls in the dimension, while the best theoretical bound is cubic. Moreover, our algorithm can be easily generalized to any convex body. We offer an open-source, optimized C++ implementation, and analyze its performance. Our code tackles problems intractable so far, offering the first efficient algorithm for zonotopes which scales to high dimensions (e.g. one hundred dimensions in less than 1 h).
For the entire collection see [Zbl 1496.68012].Slack ideals in Macaulay2https://zbmath.org/1503.520062023-03-23T18:28:47.107421Z"Macchia, Antonio"https://zbmath.org/authors/?q=ai:macchia.antonio"Wiebe, Amy"https://zbmath.org/authors/?q=ai:wiebe.amySummary: Recently Gouveia, Thomas and the authors [\textit{J. Gouveia} et al., SIAM J. Discrete Math. 33, No. 3, 1637--1653 (2019; Zbl 1423.52032)] introduced the slack realization space, a new model for the realization space of a polytope. It represents each polytope by its slack matrix, the matrix obtained by evaluating each facet inequality at each vertex. Unlike the classical model, the slack model naturally mods out projective transformations. It is inherently algebraic, arising as the positive part of a variety of a saturated determinantal ideal, and provides a new computational tool to study classical realizability problems for polytopes. We introduce the package \texttt{SlackIdeals} for \textit{Macaulay2}, that provides methods for creating and manipulating slack matrices and slack ideals of convex polytopes and matroids. Slack ideals are often difficult to compute. To improve the power of the slack model, we develop two strategies to simplify computations: we scale as many entries of the slack matrix as possible to one; we then obtain a reduced slack model combining the slack variety with the more compact Grassmannian realization space model. This allows us to study slack ideals that were previously out of computational reach. As applications, we show that the well-known Perles polytope does not admit rational realizations and prove the non-realizability of a large simplicial sphere.
For the entire collection see [Zbl 1496.68012].Line transversals in families of connected sets in the planehttps://zbmath.org/1503.520072023-03-23T18:28:47.107421Z"McGinnis, Daniel"https://zbmath.org/authors/?q=ai:mcginnis.daniel"Zerbib, Shira"https://zbmath.org/authors/?q=ai:zerbib.shiraSummary: We prove that if a family of compact connected sets in the plane has the property that every three members of it are intersected by a line, then there are three lines intersecting all the sets in the family. This answers a question of \textit{J. Eckhoff} [Discrete Comput. Geom. 9, No. 2, 203--214 (1993; Zbl 0772.52002)], who proved that, under the same condition, there are four lines intersecting all the sets. In fact, we prove a colorful version of this result under weakened conditions on the sets. Three sets \(A,B,C\) form a \textit{tight triple} if \(\mathrm{conv}(A\cup B)\cap\mathrm{conv}(A\cup C)\cap\mathrm{conv}(B\cap C)\neq\emptyset\). This notion was first introduced by Holmsen, who showed that if \(\mathcal{F}\) is a family of compact convex sets in the plane in which every three sets form a tight triple, then there is a line intersecting at least \(\frac{1}{8}|\mathcal{F}|\) members of \(\mathcal{F}\). Here we prove that if \(\mathcal{F}_1,\dots,\mathcal{F}_6\) are families of compact connected sets in the plane such that every three sets, chosen from three distinct families \(\mathcal{F}_i\), form a tight triple, then there exists \(1\leq j\leq 6\) and three lines intersecting every member of \(\mathcal{F}_j\). In particular, this improves \(\frac{1}{8}\) to \(\frac{1}{3}\) in Holmsen's result.Barycentric cuts through a convex bodyhttps://zbmath.org/1503.520082023-03-23T18:28:47.107421Z"Patáková, Zuzana"https://zbmath.org/authors/?q=ai:patakova.zuzana"Tancer, Martin"https://zbmath.org/authors/?q=ai:tancer.martin"Wagner, Uli"https://zbmath.org/authors/?q=ai:wagner.uliSummary: Let \(K\) be a convex body in \(\mathbb{R}^n\) (i.e., a compact convex set with nonempty interior). Given a point \(p\) in the interior of \(K\), a hyperplane \(h\) passing through \(p\) is called \textit{barycentric} if \(p\) is the barycenter of \(K \cap h\). In 1961, Grünbaum raised the question whether, for every \(K\), there exists an interior point \(p\) through which there are at least \(n+1\) distinct barycentric hyperplanes. Two years later, this was seemingly resolved affirmatively by showing that this is the case if \(p=p_0\) is the point of maximal \textit{depth} in \(K\). However, while working on a related question, we noticed that one of the auxiliary claims in the proof is incorrect. Here, we provide a counterexample; this re-opens Grünbaum's question. It follows from known results that for \(n \ge 2\), there are always at least three distinct barycentric cuts through the point \(p_0 \in K\) of maximal depth. Using tools related to Morse theory we are able to improve this bound: four distinct barycentric cuts through \(p_0\) are guaranteed if \(n \ge 3\).Helly-type problemshttps://zbmath.org/1503.520092023-03-23T18:28:47.107421Z"Bárány, Imre"https://zbmath.org/authors/?q=ai:barany.imre"Kalai, Gil"https://zbmath.org/authors/?q=ai:kalai.gilThe paper presents various refinements and generalizations of the theorems of Helly, Radon, Carathéodory, and Tverberg, exploring the connections of these fundamental results of convex geometry with topology. Each topic is accompanied by known conjectures and new open problems.
Reviewer: Mircea Balaj (Oradea)Optimal colored Tverberg theorems for prime powershttps://zbmath.org/1503.520102023-03-23T18:28:47.107421Z"Jojić, Duško"https://zbmath.org/authors/?q=ai:jojic.dusko"Panina, Gaiane"https://zbmath.org/authors/?q=ai:panina.gaiane-yu"Živaljević, Rade"https://zbmath.org/authors/?q=ai:zivaljevic.rade-tSummary: The colored Tverberg theorem of Blagojević Matschke, and Ziegler [\textit{P. V. M. Blagojević} et al., J. Eur. Math. Soc. (JEMS) 17, No. 4, 739--754 (2015; Zbl 1327.52009), Theorem 1.4] provides optimal bounds for the colored Tverberg problem, under the condition that the number of intersecting rainbow simplices \(r=p\) is a prime number.
Our Theorem 1.6 extends this result to an optimal colored Tverberg theorem for multisets of colored points, which is valid for each prime power \(r=p^k\), and includes Theorem 1.4 as a special case for \(k=1\). One of the principal new ideas is to replace the ambient simplex \(\Delta^N\), used in the original Tverberg theorem, by an ``abridged simplex'' of smaller dimension, and to compensate for this reduction by allowing vertices to repeatedly appear a controlled number of times in different rainbow simplices. Configuration spaces, used in the proof, are combinatorial pseudomanifolds which can be represented as multiple chessboard complexes. Our main topological tool is the Eilenberg-Krasnoselskii theory of degrees of equivariant maps for non-free actions.
A quite different generalization arises if we consider colored classes that are (approximately) two times smaller than in the classical colored Tverberg theorem. Theorem 1.8, which unifies and extends some earlier results of this type, is based on the constraint method and uses the high connectivity of the configuration space.Some new positions of maximal volume of convex bodieshttps://zbmath.org/1503.520112023-03-23T18:28:47.107421Z"Artstein-Avidan, Shiri"https://zbmath.org/authors/?q=ai:artstein-avidan.shiri"Putterman, Eli"https://zbmath.org/authors/?q=ai:putterman.eliSummary: In this paper, we extend and generalize several previous works on maximal volume positions of convex bodies. First, we analyze the maximal positive-definite image of one convex body inside another, and the resulting decomposition of the identity. We discuss continuity and differentiability of the mapping associating a body with its positive John position. We then introduce the saddle-John position of one body inside another, proving that it shares some of the properties possessed by the position of maximal volume, and explain how this can be used to improve volume ratio estimates. We investigate several examples in detail and compare these positions. Finally, we discuss the maximal intersection position of one body with respect to another, and show the existence of a natural decomposition of identity associated to this position, extending previous work which treated the case when one of the bodies is the Euclidean ball.The functional inequality for the mixed quermassintegralhttps://zbmath.org/1503.520122023-03-23T18:28:47.107421Z"Chen, Fangwei"https://zbmath.org/authors/?q=ai:chen.fangwei"Fang, Jianbo"https://zbmath.org/authors/?q=ai:fang.jianbo"Luo, Miao"https://zbmath.org/authors/?q=ai:luo.miao"Yang, Congli"https://zbmath.org/authors/?q=ai:yang.congliSummary: In this paper, the functional Quermassintegrals of a log-concave function in \(\mathbb{R}^n\) are discussed. The functional inequality for the \(i\)th mixed Quermassintegral is established. Moreover, as a special case, a weaker log-Quermassintegral inequality in \(\mathbb{R}^n\) is obtained.Geometric estimates in interpolation on an \(n\)-dimensional ballhttps://zbmath.org/1503.520132023-03-23T18:28:47.107421Z"Nevskiĭ, Mikhail Viktorovich"https://zbmath.org/authors/?q=ai:nevskii.mikhail-viktorovichSummary: Suppose \(n\in \mathbb{N}\). Let \(B_n\) be a Euclidean unit ball in \(\mathbb{R}^n\) given by the inequality \(\|x\|\leq 1\), \(\|x\|:=\bigg(\sum\limits_{i=1}^n x_i^2\bigg)^{\frac{1}{2}} \). By \(C(B_n)\) we mean a set of continuous functions \(f:B_n\to\mathbb{R}\) with the norm \(\|f\|_{C(B_n)}:=\max\limits_{x\in B_{n}}|f(x)|\). The symbol \(\Pi_1\) (\(\mathbb{R}^n\)) denotes a set of polynomials in \(n\) variables of degree \(\leq 1\), i. e., linear functions upon \(\mathbb{R}^n\). Assume that \(x^{(1)}, \ldots, x^{(n+1)}\) are vertices of an \(n\)-dimensional nondegenerate simplex \(S\subset B_n\). The interpolation projector \(P:C(B_n)\to \Pi_1(\mathbb{R}^n)\) corresponding to \(S\) is defined by the equalities \(P\,f\,(x^{(j)})=f\,(x^{(j)})\). Denote by \(\|P\|_{B_n}\) the norm of \(P\) as an operator from \(C(B_n)\) onto \(C(B_n)\). Let us define \(\theta_n(B_n)\) as the minimal value of \(\|P\|_{B_n}\) under the condition \(x^{(j)}\in B_n\). We describe the approach in which the norm of the projector can be estimated from the bottom through the volume of the simplex. Let \(\chi_n(t):=\frac{1}{2^{n}n!}\left[ (t^2-1)^n \right]^{(n)}\) be the standardized Legendre polynomial of degree \(n\). We prove that \(\|P\|_{B_n} \geq \chi_n^{-1} \bigg(\frac{\text{vol}(B_n)}{\text{vol}(S)}\bigg).\) From this, we obtain the equivalence \(\theta_n(B_n)\asymp \sqrt{n}\). Also we estimate the constants from such inequalities and give the comparison with the similar relations for linear interpolation upon the \(n\)-dimensional unit cube. These results have applications in polynomial interpolation and computational geometry.The Orlicz Minkowski problem for the electrostatic \(\mathfrak{p}\)-capacityhttps://zbmath.org/1503.520142023-03-23T18:28:47.107421Z"Xiong, Ge"https://zbmath.org/authors/?q=ai:xiong.ge"Xiong, Jiawei"https://zbmath.org/authors/?q=ai:xiong.jiaweiIn this paper, the authors prove the existence of solutions to the Orlicz Minkowski problem for \(p\)-capacity with \(p>n\). More precisely, the authors consider the necessary and sufficient conditions on a function \(\varphi\) and a finite Borel measure \(\mu\) on \(\mathbb{S}^{n-1}\) so that there exists a convex body \(K\) with
\[
\alpha d\mu_p(K,\cdot)=\frac{d\mu}{\varphi(h_K)},
\]
where \(\mu_p(K,\cdot)\) is the electrostatic \(p\)-capacitary measure of \(K\), \(h_K\) is the support function of \(K\), and \(\alpha>0\) is a constant. The main result of the paper shows that if \(\mu\) is not concentrated on any closed hemisphere, \(p>n\), \(\varphi:(0,\infty)\rightarrow(0,\infty)\) is continuous, decreasing, and \(\varphi(s)\rightarrow\infty\) as \(s\rightarrow 0+\), then the existence of such a convex body and \(\alpha>0\) is guaranteed.
Reviewer: Mariana Vega Smit (Bellingham)Approximation of spherical bodies of constant width and reduced bodieshttps://zbmath.org/1503.520152023-03-23T18:28:47.107421Z"Lassak, Marek"https://zbmath.org/authors/?q=ai:lassak.marekA theorem by \textit{W. Blaschke} [Math. Ann. 76, 504--513 (1915; JFM 45.0731.04)] says that for every convex body of constant width \(w\) in the Euclidean plane \(E^2\) and every \(\varepsilon>0\) there exists a convex body of constant width \(w\) whose boundary consists only of pieces of circles of radius \(w\) such that the Hausdorff distance between the two bodies is at most \(\varepsilon\). A generalization of this fact for normed planes was given by the author [J. Convex Anal. 19, No. 3, 865--874 (2012; Zbl 1261.52004)], where also the approximation of reduced bodies was considered. A corollary (Corollary 4.2) of the present paper is the analog of this theorem for bodies of constant width on the unit sphere \(S^2\) of the three-dimensional Euclidean space \(E^3\), while Theorem 4.1 gives a general version for reduced convex bodies on \(S^2\).
Reviewer: T. D. Narang (Amritsar)New proofs for some results on spherically convex setshttps://zbmath.org/1503.520162023-03-23T18:28:47.107421Z"Zalinescu, Constantin"https://zbmath.org/authors/?q=ai:zalinescu.constantinSummary: In [J. Convex Anal. 28, No. 1, 103--122 (2021; Zbl 1455.52007)], \textit{Q. Guo} and \textit{Y. Peng} define the notions of spherical convex sets and functions on ``general curved surfaces'' in \(\mathbb{R}^n\) \((n \geq 2)\), they study several properties of these classes of sets and functions, and they establish analogues of Radon, Helly, Carathéodory and Minkowski theorems for spherical convex sets, as well as some properties of spherical convex functions which are analogous to those of usual convex functions. In obtaining such results, the authors use an analytic approach based on their definitions. Our aim in this note is to provide simpler proofs for these results on spherical convex sets; our proofs are based on some characterizations/representations of spherical convex sets by usual convex sets in \(\mathbb{R}^n\). Moreover, we provide a new proof of the recent result of \textit{H. Han} and \textit{T. Nashimura} on the separation of spherical convex sets established in [``Spherical separation theorem'', Preprint, \url{arXiv:2002.06558}]. Our proof is based on a result stated in locally convex spaces.Partial permutation and alternating sign matrix polytopeshttps://zbmath.org/1503.520172023-03-23T18:28:47.107421Z"Heuer, Dylan"https://zbmath.org/authors/?q=ai:heuer.dylan"Striker, Jessica"https://zbmath.org/authors/?q=ai:striker.jessicaSummary: We define and study a new family of polytopes which are formed as convex hulls of partial alternating sign matrices. We determine the inequality descriptions, number of facets, and face lattices of these polytopes. We also study partial permutohedra that we show arise naturally as projections of these polytopes. We enumerate facets and also characterize the face lattices of partial permutohedra in terms of chains in the Boolean lattice. Finally, we have a result and a conjecture on the volume of partial permutohedra when one parameter is fixed to be two.The lower bound theorem for \(d\)-polytopes with \(2d+1\) verticeshttps://zbmath.org/1503.520182023-03-23T18:28:47.107421Z"Pineda-Villavicencio, Guillermo"https://zbmath.org/authors/?q=ai:pineda-villavicencio.guillermo"Yost, David"https://zbmath.org/authors/?q=ai:yost.david-tExact lower bounds for the number of \(k\)-faces of \(d\)-polytopes (not necessarily simple or simplicial) with \(2d+1\) vertices are established, and the minimizers are characterized. All \(d\)-polytopes with \(d + 3\) vertices and only one or two edges more than the minimum are characterized.
Reviewer: Gaiane Panina (Sankt-Peterburg)Linear interpolation on a Euclidean ball in \(\mathbb{R}^n\)https://zbmath.org/1503.520192023-03-23T18:28:47.107421Z"Nevskiĭ, Mikhail Viktorovich"https://zbmath.org/authors/?q=ai:nevskii.mikhail-viktorovich"Ukhalov, Alekseĭ Yur'evich"https://zbmath.org/authors/?q=ai:ukhalov.aleksei-yurevichSummary: For \(x^{(0)}\in\mathbb{R}^n\), \(R>0\), by \(B=B(x^{(0)};R)\) we denote a Euclidean ball in \(\mathbb{R}^n\) given by the inequality \(\|x-x^{(0)}\|\leq R\), \(\|x\|:=(\sum_{i=1}^n x_i^2)^{1/2}\). Put \(B_n:=B(0,1)\). We mean by \(C(B)\) the space of continuous functions \(f:B\to\mathbb{R}\) with the norm \(\|f\|_{C(B)}:=\max_{x\in B}|f(x)|\) and by \(\Pi_1(\mathbb{R}^n)\) the set of polynomials in \(n\) variables of degree \(\leq 1\), i. e. linear functions on \(\mathbb{R}^n\). Let \(x^{(1)}, \ldots, x^{(n+1)}\) be the vertices of \(n\)-dimensional nondegenerate simplex \(S\subset B\). The interpolation projector \(P:C(B)\to \Pi_1(\mathbb{R}^n)\) corresponding to \(S\) is defined by the equalities \(P\,f\,(x^{(j)}=f\,(x^{(j)})\). Denote by \(\|P\|_B\) the norm of \(P\) as an operator from \(C(B)\) into \(C(B)\). Let us define \(\theta_n(B)\) as minimal value of \(\|P\|_B\) under the condition \(x^{(j)}\in B\). In the paper, we obtain the formula to compute \(\|P\|_B\) making use of \(x^{(0)}, R\), and coefficients of basic Lagrange polynomials of \(S\). In more details we study the case when \(S\) is a regular simplex inscribed into \(B_n\). In this situation, we prove that \(\|P\|_{B_n}=\max\{\psi(a),\psi(a+1)\},\) where \(\psi(t)=\frac{2\sqrt{n}}{n+1}\bigl(t(n+1-t)\bigr)^{1/2}+ \bigl|1-\frac{2t}{n+1}\bigr| (0\leq t\leq n+1)\) and integer \(a\) has the form \(a=\bigl\lfloor\frac{n+1}{2}-\frac{\sqrt{n+1}}{2}\bigr\rfloor.\) For this projector, \( \sqrt{n}\leq\|P\|_{B_n}\leq\sqrt{n+1} \). The equality \(\|P\|_{B_n}=\sqrt{n+1}\) takes place if and only if \(\sqrt{n+1}\) is an integer number. We give the precise values of \(\theta_n(B_n)\) for \(1\leq n\leq 4\). To supplement theoretical results we present computational data. We also discuss some other questions concerning interpolation on a Euclidean ball.Difference between families of weakly and strongly maximal integral lattice-free polytopeshttps://zbmath.org/1503.520202023-03-23T18:28:47.107421Z"Averkov, Gennadiy"https://zbmath.org/authors/?q=ai:averkov.gennadiySummary: A \(d\)-dimensional closed convex set \(K\) in \(\mathbb{R}^d\) is said to be lattice-free if the interior of \(K\) is disjoint with \(\mathbb{Z}^d\). We consider the following two families of lattice-free polytopes: the family \(\mathcal{L}^d\) of integral lattice-free polytopes in \(\mathbb{R}^d\) that are not properly contained in another integral lattice-free polytope and its subfamily \(\mathcal{M}^d\) consisting of integral lattice-free polytopes in \(\mathbb{R}^d\) which are not properly contained in another lattice-free set. It is known that \(\mathcal{M}^d = \mathcal{L}^d\) holds for \(d \le 3\) and, for each \(d \ge 4\), \(\mathcal{M}^d\) is a proper subfamily of \(\mathcal{L}^d\). We derive a super-exponential lower bound on the number of polytopes in \(\mathcal{L}^d {\setminus} \mathcal{M}^d\) (with standard identification of integral polytopes up to affine unimodular transformations).
For the entire collection see [Zbl 1497.52003].The characterisation problem of Ehrhart polynomials of lattice polytopeshttps://zbmath.org/1503.520212023-03-23T18:28:47.107421Z"Higashitani, Akihiro"https://zbmath.org/authors/?q=ai:higashitani.akihiroSummary: One of the most important invariants of a lattice polytope is the Ehrhart polynomial. The problem of which polynomials can be Ehrhart polynomials of lattice polytopes is now well-studied. In this survey paper, after recalling the fundamental properties of the Ehrhart polynomials of lattice polytopes, we survey the results on this problem including recent developments. We discuss the characterisation of Ehrhart polynomials in several particular cases: small dimensions; small volumes; palindromic; small degrees. We also suggest some possible further questions.
For the entire collection see [Zbl 1497.52003].Linear recursions for integer point transformshttps://zbmath.org/1503.520222023-03-23T18:28:47.107421Z"Jochemko, Katharina"https://zbmath.org/authors/?q=ai:jochemko.katharinaSummary: We consider the integer point transform
\[
\sigma_P (\mathbf{x}) = \sum_{\mathbf{m}\in P\cap \mathbb{Z}^n} \mathbf{x}^\mathbf{m}\in \mathbb{C}[x_1^{\pm 1},\dots, x_n^{\pm 1}]
\]
of a polytope \(P\subset \mathbb{R}^n\). We show that if \(P\) is a lattice polytope then for any polytope \(Q\) the sequence \(\lbrace \sigma_{kP+Q}(\mathbf{x})\rbrace_{k\ge 0}\) satisfies a multivariate linear recursion that only depends on the vertices of \(P\). We recover Brion's Theorem and by applying our results to Schur polynomials we disprove a conjecture of \textit{P. Alexandersson} [J. Comb. Theory, Ser. A 122, 1--8 (2014; Zbl 1311.05205)].
For the entire collection see [Zbl 1497.52003].Symmetric edge polytopes and matching generating polynomialshttps://zbmath.org/1503.520232023-03-23T18:28:47.107421Z"Ohsugi, Hidefumi"https://zbmath.org/authors/?q=ai:ohsugi.hidefumi"Tsuchiya, Akiyoshi"https://zbmath.org/authors/?q=ai:tsuchiya.akiyoshiSummary: Symmetric edge polytopes \(\mathcal{A}_G\) of type A are lattice polytopes arising from the root system \(A_n\) and finite simple graphs \(G\). There is a connection between \(\mathcal{A}_G\) and the Kuramoto synchronization model in physics. In particular, the normalized volume of \(\mathcal{A}_G\) plays a central role. In the present paper, we focus on a particular class of graphs. In fact, for any cactus graph \(G\), we give a formula for the \(h^*\)-polynomial of \(\mathcal{A}_{\widehat{G}}\) by using matching generating polynomials, where \(\widehat{G}\) is the suspension of \(G\). This gives also a formula for the normalized volume of \(\mathcal{A}_{\widehat{G}}\). Moreover, via methods from chemical graph theory, we show that for any cactus graph \(G\), the \(h^*\)-polynomial of \(\mathcal{A}_{\widehat{G}}\) is real-rooted. Finally, we extend the discussion to symmetric edge polytopes of type \(B\), which are lattice polytopes arising from the root system \(B_n\) and finite simple graphs.The reflexive dimension of (0, 1)-polytopeshttps://zbmath.org/1503.520242023-03-23T18:28:47.107421Z"Tsuchiya, Akiyoshi"https://zbmath.org/authors/?q=ai:tsuchiya.akiyoshiSummary: Haase and Melnikov showed that every lattice polytope is unimodularly equivalent to a face of some reflexive polytope. The reflexive dimension of a lattice polytope \(P\) is the minimal \(d\) so that \(P\) is unimodularly equivalent to a face of some \(d\)-dimensional reflexive polytope. Computing the reflexive dimension of a lattice polytope is a hard problem in general. In this survey, we discuss the reflexive dimension of a (0, 1)-polytope. In particular, virtue of the algebraic technique on Gröbner bases and a linear algebraic technique, many families of reflexive polytopes arising from several classes of (0, 1)-polytopes are presented, and we see that the (0, 1)-polytopes are unimodularly equivalent to facets of some reflexive polytopes.
For the entire collection see [Zbl 1497.52003].Tiling the plane with hexagons: improved separations for \(k\)-colouringshttps://zbmath.org/1503.520252023-03-23T18:28:47.107421Z"de Grey, Aubrey D. N. J."https://zbmath.org/authors/?q=ai:de-grey.aubrey-d-n-j"Parts, Jaan"https://zbmath.org/authors/?q=ai:parts.jaanIt has been well-known that given a tiling of the plane with regular hexagons of diameter 1 (more properly, infinitesimally less than 1), the hexagons can be colored with 7 colors, such that no unit distance occurs between points of identically colored hexagons. Furthermore, there is a small \textit{interval}, namely \([1, \sqrt{7}/2)\), such that numbers from this interval do not occur as distances between points of identically colored hexagons. A similar question can be asked for \(k\geq 7\)-colorations of the hexagons: what is the largest possible \(d(k)\), such that for some \(k\)-coloration of the hexagons, no number from the interval \([1,d(k))\) appears as a distance between points of identically colored hexagons. For \(k>7\) the exact value of \(d(k)\) is not known, although estimates exists. The paper under review studies tilings with congruent, but not regular hexagons of diameter 1, and achieves somewhat bigger intervals, whose elements do not occur as distances between points of identically colored hexagons, than what was achieved for colorations of tilings with regular hexagons of diameter 1.
Reviewer: László A. Székely (Columbia)On the dimension of the \(k\)-medial axis for an arbitrary closed sethttps://zbmath.org/1503.520262023-03-23T18:28:47.107421Z"Liang, Xiangyu"https://zbmath.org/authors/?q=ai:liang.xiangyuThe \(k\)-medial axis \(M_k\) of a closed set \(E \subseteq {\mathbb R}^n\) consists of all points in \({\mathbb R}^n\) that have at least \(k\) nearest points in \(E\) in general position (i.e.\ not all in an affine subspace of dimension \(k-2\)). The author proves for \(1\leq k\leq n+1\) that \(M_k\) is \((n+k-1)\)-rectifiable; in particular, \(M_k\) has Hausdorff dimension at most \(n+k-1\). This is related to a conjecture of \textit{P. Erdős} [Bull. Am. Math. Soc. 51, 728--731 (1945; Zbl 0063.01269)].
Reviewer: Theo Grundhöfer (Würzburg)Many touchings force many crossingshttps://zbmath.org/1503.520272023-03-23T18:28:47.107421Z"Pach, János"https://zbmath.org/authors/?q=ai:pach.janos"Tóth, Géza"https://zbmath.org/authors/?q=ai:toth.gezaSummary: Given \(n\) continuous open curves in the plane, we say that a pair is touching if they have only one interior point in common and at this point the first curve does not get from one side of the second curve to its other side. Otherwise, if the two curves intersect, they are said to form a crossing pair. Let \(t\) and \(c\) denote the number of touching pairs and crossing pairs, respectively. We prove that \(c\geq\frac{1}{10^5}\frac{t^2}{n^2}\), provided that \(t\geq 10n\). Apart from the values of the constants, this result is best possible.
For the entire collection see [Zbl 1381.68007].Maximal tilings with the minimal tile propertyhttps://zbmath.org/1503.520282023-03-23T18:28:47.107421Z"Praton, Iwan"https://zbmath.org/authors/?q=ai:praton.iwanLet us be given a tiling of the unit square with squares of side lengths \(s_1,s_2,\dots, s_n\). How small can be \(s_1+s_2+\dots+s_n\)?
This question can be attributed to Erdős and Soifer. J. Alm suggested a more restricted (and therefore hopefully more tractable) version of this problem: consider only those tilings, where the smallest tile can tile all larger tiles. The nomenclature for such tilings is that they have the \textit{Minimal Tile Property} (MTP). Denote the minimum sum by \(f(n)\) and \(f_M(n)\), respectively. The paper studies the conjecture \(f_M(k+3)=k+ 1/k\). This conjecture is based on the construction of the standard tiling of the unit square into \(k^2\) squares of edge length \(1/k\), and then subdividing one small square into four congruents squares. The main result of the paper is that if this conjecture fails, then only one copy of the smallest square is present in the tiling, and at least 3 different tile sizes are present.
Reviewer: László A. Székely (Columbia)Parallel packing squares into an obtuse trianglehttps://zbmath.org/1503.520292023-03-23T18:28:47.107421Z"Januszewski, Janusz"https://zbmath.org/authors/?q=ai:januszewski.janusz"Liu, Xi"https://zbmath.org/authors/?q=ai:liu.xi"Su, Zhanjun"https://zbmath.org/authors/?q=ai:su.zhanjun"Zielonka, Łukasz"https://zbmath.org/authors/?q=ai:zielonka.lukaszSummary: Suppose that \(T(\alpha, \beta)\) is an obtuse triangle with base length 1 and with base angles measuring \(\alpha\) and \(\beta\) (where \(\alpha >90^{\circ})\). Let \(S\) be a square with a side parallel to the base of \(T(\alpha, \beta)\) and let \(\{S_i\}\) be a collection of the homothetic copies of \(S\). In this note a tight upper bound of the sum of the areas of squares from \(\{S_i\}\) that can be parallel packed into \(T(\alpha, \beta)\) is given. This result complements the previous upper bound obtained for \(\alpha \le 90^{\circ}\).Cohomology groups for spaces of twelve-fold tilingshttps://zbmath.org/1503.520302023-03-23T18:28:47.107421Z"Bédaride, Nicolas"https://zbmath.org/authors/?q=ai:bedaride.nicolas"Gähler, Franz"https://zbmath.org/authors/?q=ai:gahler.franz"Lecuona, Ana G."https://zbmath.org/authors/?q=ai:lecuona.ana-gIn this paper, the authors compute the cohomology of some planar tiling spaces, enhancing their understanding of them and providing the community with some complete calculations. They focus on a two-parameter family of cut-and-project tiling spaces: the generalized twelve-fold tilings, introducing new effective techniques to compute the cohomology groups of these tilings.
Reviewer: Altino Manuel Folgado dos Santos (Vila Real)Delone sets and tilings: local approachhttps://zbmath.org/1503.520312023-03-23T18:28:47.107421Z"Dolbilin, N. P."https://zbmath.org/authors/?q=ai:dolbilin.nikolai-p"Shtogrin, M. I."https://zbmath.org/authors/?q=ai:shtogrin.mikhail-ivanovichSummary: We present new results in the local theory of Delone sets, regular systems, and isogonal tilings. In particular, we prove a local criterion for isogonal tilings of the Euclidean space. This criterion is then applied to the study of \(2R\)-isometric Delone sets, where \(R\) is the covering radius for these sets. For regular systems in the plane we establish the exact value \(\widehat{\rho}_2=4R\) of the regularity radius. We prove that in any cell of the Delone tiling in an arbitrary Delone set in the plane, there is a vertex at which the local group is crystallographic. Hence, the subset of points with local crystallographic groups in a Delone set in the plane is itself a Delone set with covering radius at most \(2R\).Enriques surfaces and an Apollonian packing in eight dimensionshttps://zbmath.org/1503.520322023-03-23T18:28:47.107421Z"Baragar, Arthur"https://zbmath.org/authors/?q=ai:baragar.arthurSummary: We call a packing of hyperspheres in \(n\) dimensions an Apollonian sphere packing if the spheres intersect tangentially or not at all; they fill the \(n\)-dimensional Euclidean space; and every sphere in the packing is a member of a cluster of \(n+2\) mutually tangent spheres (and a few more properties described herein). In this paper, we describe an Apollonian packing in eight dimensions that naturally arises from the study of generic nodal Enriques surfaces. The \(E_7, E_8\) and Reye lattices play roles. We use the packing to generate an Apollonian packing in nine dimensions, and a cross section in seven dimensions that is weakly Apollonian. Maxwell described all three packings but seemed unaware that they are Apollonian. The packings in seven and eight dimensions are different than those found in an earlier paper. In passing, we give a sufficient condition for a Coxeter graph to generate mutually tangent spheres and use this to identify an Apollonian sphere packing in three dimensions that is not the Soddy sphere packing.Affine dimers from characteristic polygonshttps://zbmath.org/1503.520332023-03-23T18:28:47.107421Z"Holmes, Daniel"https://zbmath.org/authors/?q=ai:holmes.danielThis paper mainly builds on [\textit{J. Forsgård} and \textit{P. Johansson}, Math. Z. 278, No. 1--2, 25--38 (2014; Zbl 1328.14097); \textit{J. Forsgård}, Ann. Inst. Henri Poincaré D, Comb. Phys. Interact. (AIHPD) 6, No. 2, 199--219 (2019; Zbl 1417.82009)].
The main objects of study in this paper are \textit{dimer models} or \textit{dimers}, i.e., bipartite multigraphs \(G\) embedded on the torus \(\mathbb{T}^2\). A line on \(\mathbb{T}^2\) is defined by the image of a line in \(\mathbb{R}^2\) under the quotient map \(q\colon\mathbb{R}^2\longrightarrow\mathbb{R}^2/\mathbb{Z}^2\simeq\mathbb{T}^2\). Lines in this paper are assumed to be \textit{closed geodesics}, i.e., closed curves in \(\mathbb{T}^2\). If such a line \(H\) is given an orientation, its direction can be defined by two unique coprime integers, \(a,b\in\mathbb{Z}\), which form the \textit{homology class} \([H]:=(a,b)\in\mathbb{Z}^2\) of \(H\). An \textit{affine dimer} is a dimer which arises from a special type of orientable line arrangements on \(\mathbb{T}^2\) called \textit{admissible oriented line arrangements}. Every affine dimer admits a lattice polygon called \textit{homology polygon}, which is uniquely defined by the homology classes of the lines of its associated admissible oriented line arrangement.
Conversely, every lattice polygon yields multisets of homology classes, that can be interpreted (albeit not uniquely) as oriented line arrangements. However, it is not true that for any lattice polygon, one of the oriented line arrangements it yields is necessarily admissible. Hence, the goal of this paper is to classify which convex polygons appear as homology polytopes of an affine dimer. A partial result is presented in form of a necessary (Theorem A) and a sufficient (Theorem B) condition on lattice polygons. In particular, among other things, lattice triangles, reflexive polygons, and polygons with two interior lattice points are confirmed to admit affine dimers.
Reviewer: Max Kölbl (Osaka)On generalized Minkowski arrangementshttps://zbmath.org/1503.520342023-03-23T18:28:47.107421Z"Kadlicskó, Máté"https://zbmath.org/authors/?q=ai:kadlicsko.mate"Lángi, Zsolt"https://zbmath.org/authors/?q=ai:langi.zsoltSummary: The concept of a Minkowski arrangement was introduced by \textit{L. Fejes Tóth} in [Proc. Am. Math. Soc. 16, 999--1004 (1965; Zbl 0129.37404)] as a family of centrally symmetric convex bodies with the property that no member of the family contains the center of any other member in its interior. This notion was generalized by \textit{L. Fejes Tóth} in [Math. Ann. 171, 97--103 (1967; Zbl 0145.42702)], who called a family of centrally symmetric convex bodies a generalized Minkowski arrangement of order \(\mu\) for some \(0 < \mu < 1\) if no member \(K\) of the family overlaps the homothetic copy of any other member \(K'\) with ratio \(\mu\) and with the same center as \(K'\). In this note we prove a sharp upper bound on the total area of the elements of a generalized Minkowski arrangement of order \(\mu\) of finitely many circular disks in the Euclidean plane. This result is a common generalization of a similar result of Fejes Tóth for Minkowski arrangements of circular disks, and a result of Böröczky and Szabó about the maximum density of a generalized Minkowski arrangement of circular disks in the plane. In addition, we give a sharp upper bound on the density of a generalized Minkowski arrangement of homothetic copies of a centrally symmetric convex body.An almost optimal bound on the number of intersections of two simple polygonshttps://zbmath.org/1503.520352023-03-23T18:28:47.107421Z"Ackerman, Eyal"https://zbmath.org/authors/?q=ai:ackerman.eyal"Keszegh, Balázs"https://zbmath.org/authors/?q=ai:keszegh.balazs"Rote, Günter"https://zbmath.org/authors/?q=ai:rote.gunterSummary: What is the maximum number of intersections of the boundaries of a simple \(m\)-gon and a simple \(n\)-gon? This is a basic question in combinatorial geometry, and the answer is easy if at least one of \(m\) and \(n\) is even: If both \(m\) and \(n\) are even, then every pair of sides may cross and so the answer is \textit{mn}. If exactly one polygon, say the \(n\)-gon, has an odd number of sides, it can intersect each side of the \(m\)-gon polygon at most \(n-1\) times; hence there are at most \(mn-m\) intersections. It is not hard to construct examples that meet these bounds. If both \(m\) and \(n\) are odd, the best known construction has \(mn-(m+n)+3\) intersections, and it is conjectured that this is the maximum. However, the best known upper bound is only \(mn-(m + \lceil{n}/{6} \rceil )\), for \(m \ge n\). We prove a new upper bound of \(mn-(m+n)+C\) for some constant \(C\), which is optimal apart from the value of \(C\).Arrangements of pseudocircles: triangles and drawingshttps://zbmath.org/1503.520362023-03-23T18:28:47.107421Z"Felsner, Stefan"https://zbmath.org/authors/?q=ai:felsner.stefan"Scheucher, Manfred"https://zbmath.org/authors/?q=ai:scheucher.manfredSummary: A pseudocircle is a simple closed curve on the sphere or in the plane. The study of arrangements of pseudocircles was initiated by Grünbaum, who defined them as collections of simple closed curves that pairwise intersect in exactly two crossings. Grünbaum conjectured that the number of triangular cells \(p_3\) in digon-free arrangements of \(n\) pairwise intersecting pseudocircles is at least \(2n-4\). We present examples to disprove this conjecture. With a recursive construction based on an example with 12 pseudocircles and 16 triangles we obtain a family with \(p_3(\mathscr {A})/n \rightarrow 16/11 = 1.\overline{45}\). We expect that the lower bound \(p_3(\mathscr {A})\geq 4n/3\) is tight for infinitely many simple arrangements. It may however be that digon-free arrangements of \(n\) pairwise intersecting circles indeed have at least \(2n-4\) triangles.
For pairwise intersecting arrangements with digons we have a lower bound of \(p_3\geq 2n/3\), and conjecture that \(p_3\geq n-1\).
Concerning the maximum number of triangles in pairwise intersecting arrangements of pseudocircles, we show that \(p_3\leq 2n^2/3 +O(n)\). This is essentially best possible because families of pairwise intersecting arrangements of \(n\) pseudocircles with \(p_3/n^2\rightarrow 2/3\) as \(n\rightarrow\infty\) are known.
The paper contains many drawings of arrangements of pseudocircles and a good fraction of these drawings was produced automatically from the combinatorial data produced by the generation algorithm. In the final section we describe some aspects of the drawing algorithm.
For the entire collection see [Zbl 1381.68007].Generalized virtual polytopes and quasitoric manifoldshttps://zbmath.org/1503.520372023-03-23T18:28:47.107421Z"Limonchenko, Ivan Yu."https://zbmath.org/authors/?q=ai:limonchenko.ivan-yu"Monin, Leonid V."https://zbmath.org/authors/?q=ai:monin.leonid-v"Khovanskii, Askold G."https://zbmath.org/authors/?q=ai:khovanskii.askold-gThe paper under review is devoted to generalized virtual polytopes based on the study of topological properties of affine subspace arrangements in real spaces.
Let \(Q\) be a polynomial of degree at most \(k\) (homogeneous polynomial of degree \(k\)) on \(\mathbb{R}^{n}\), and denote by \(\omega = dx_{1} \wedge \dots \wedge dx_{n}\) the standard volume form on \(\mathbb{R}^{n}\). Let \(C_{s}\) be the cone of strictly convex bodies \(\triangle \subset \mathbb{R}^{n}\) with a smooth boundary. Then the function
\[
F(\triangle) = \int_{\triangle}Q_{\omega}
\]
on the cone \(C_{s}\) is a polynomial of degree at most \(k+n\) (homogeneous polynomial of degree \(k+n\)). Now, to extend the domain of the integral functional to the entire vector space generated by the cone \(C_{s}\), the authors introduce the notion of a virtual convex body as a formal difference of two convex bodies with the identification \(\triangle_{1} - \triangle_{2} = \triangle_{3} - \triangle_{4} \iff \triangle_{1} + \triangle_{4} = \triangle_{2}+\triangle_{3}\).
The main result provided by the authors, in the setting presented above, can be formulated as follows.
Theorem A. Let \(M\) be the space of virtual convex bodies representable as a difference of convex bodies from the cone \(C_{s}\). Then the functional \(F\) on \(C_{s}\) can be extended as an integral of the form \(Q\omega\) over the chain of virtual convex bodies. Moreover, such an extension will be a polynomial on \(M\).
Next, the authors study homological properties of unions \(X\) of (finite) arrangements of affine subspaces \(\{L_{i}\}\) in the real space \(L:=\mathbb{R}^{n}\) by means of the nerves \(K_{X}\) and their closed coverings by the sets \(L_{i}\). Consider a set \(\{L_{i}\}\) of affine subspaces in \(L\) indexed by elements \(i \in I\) and let \(X = \bigcup_{i \in I}L_{i}\) be their union. Notice that \(X\), as a topological space, has a natural covering by the affine subspaces \(L_{i}\). The nerve \(K_{X}\) of the natural covering of \(X\) is the simplicial complex with the vertex set indexed by \(I\) (one vertex for each index \(i \in I\)). A set of vertices \(v_{i_{1}}, \dots, v_{i_{k}}\) defines a simplex in \(K_{X}\) if and only if the intersection \(L_{i_{1}} \cap \dots \cap L_{i_{k}}\) is not empty. Consider another collection of affine subspaces \(\{M_{i}\}\) in a vector space \(M\) indexed by the same set \(I\), with a nerve \(K_{Y}\) corresponding to the natural covering of their union \(Y\). We say that the nerve \(K_{X}\) of the collection \(\{L_{i}\}\) dominates the nerve \(K_{Y}\) of the collection \(\{M_{i}\}\) if
\[
\bigcap_{j \in J} L_{j} \neq \emptyset \quad \Rightarrow \quad \bigcap_{j \in J} M_{j} \neq \emptyset \,\, \text{ for all }J \subset I.
\]
We write in such a case that \(K_{X}\geq K_{Y}\). We say that a continuous map \(f : X \rightarrow Y\) is compatible with \(K_{X}\) and \(K_{Y}\) if
\[
x \in L_{i_{1}} \cap \dots \cap L_{i_{k}} \Rightarrow f(x) \in M_{i_{1}} \cap \dots \cap M_{i_{k}}.
\]
In order to study homological properties of unions of affine subspace arrangements, the authors show the following result.
Theorem B.
\begin{itemize}
\item[i)] If a map \(f : X \rightarrow Y\) compatible with \(K_{X}\) and \(K_{Y}\) exists, then the condition \(K_{X}\geq K_{Y}\) holds.
\item[ii)] If a map \(f : X \rightarrow Y\) compatible with \(K_{X}\) and \(K_{Y}\) exists, then it is unique up to homotopy.
\item[iii)] If \(K_{X}\) is isomorphic to \(K_{Y}\), then the map \(f : K_{X} \rightarrow K_{Y}\) compatible with \(K_{X}\) and \(K_{Y}\) provides a homotopy equivalence between \(X\) and \(Y\).
\end{itemize}
Recall that a good triangulation of \(X = \bigcup_{i\in I} L_{i}\) is a triangulation such that the following condition holds: the vertex set of a simplex \(S\) in a good triangulation is totally ordered, i.e., there is an order on the vertex set \(\{v_{i_{1}},\dots, v_{i_{s}}\}\) of \(S\) such that \(I(v_{i_{1}}) \subset\dots \subset I(v_{i_{s}})\), where for a point \(x \in X\) by \(I(x)\) we mean the subset of indices such that \(x \in L_{i}\) if and only if \(i \in I(x)\).
Theorem C. For any finite union \(X = \bigcup_{i\in I} L_{i}\) of affine subspaces \(L_{i}\) in a linear space \(L\), one can construct a good triangulation of \(X\).
Now we pass to topological properties. Suppose that we have an arrangement of affine hyperplanes \(\{H_{i}\}\) in \(L:=\mathbb{R}^{n}\). We call it non-degenerate if there is no proper linear subspace \(V \subset \mathbb{R}^{n}\) which is parallel to all the hyperplanes \(H_{i}\). Then the union \(X\) of such an arrangement has the homotopy type of a wedge of \((n-1)\)-dimensional spheres, in which the number of spheres is equal to the number of bounded regions in \(L\setminus X\). Therefore each cycle \(\Gamma \in H_{n-1}(X,\mathbb{Z})\) can be represented as a linear combination \(\Gamma = \sum_{j} \lambda_{j} \partial \triangle_{j}\), where each coefficient \(\lambda_{j}\) equals the winding number of the cycle \(\Gamma\) around a point \(a_{J} \in \triangle_{j} \setminus \partial \triangle_{j}\). Here by \(\triangle_{j}\) we denote the closure of a bounded open polyhedron representing a bounded component of \(L \setminus X\).
We say that two hyperplane arrangements \(\mathcal{H}_{1}\) and \(\mathcal{H}_{2}\) are combinatorially equivalent if the corresponding nerves \(K_{\mathcal{H}_{1}}\) and \(K_{\mathcal{H}_{2}}\) are isomorphic.
Theorem D. Suppose that \(\mathcal{H} = \{H_{1}, \dots, H_{s}\}\) and \(\mathcal{H}' = \{H_{1}^{'}, \dots, H_{s}^{'}\}\) are two combinatorially equivalent hyperplane arrangements, and let \(X = \bigcup_{i} H_{i}\) and \(Y = \bigcup_{i}H_{i}^{'}\) be the corresponding unions of hyperplanes. Then there exists a canonical homotopy equivalence \(f : X \rightarrow Y\).
In order to study the homotopy type of a union of affine subspaces in \(\mathbb{R}^{n}\), one considers finite unions \(U \subset \mathbb{R}^{n}\) of open convex bodies \(U = \bigcup_{i} U_{i}\). Recall that a tail cone \(\mathrm{tail}(U)\) of a convex body \(U\) is the set of points \(v \in \mathbb{R}^{n}\) such that the inclusion \(a+tv \in U\) holds for any \(a \in U\) and \(t\geq 0\).
If the set \(\mathrm{tail}(U_{i})\) is a linear space \(L_{i}\), then along with \(U_{i}\) one can consider a shifted space \(a_{i} + L_{i} \subset U_{i}\), where \(a_{i}\) is an arbitrary point in \(U_{i}\).
Theorem E. Let \(U \subset \mathbb{R}^{n}\) be a finite union of open convex bodies. Then the set \(\mathbb{R}^{n} \setminus U\) is homotopy equivalent to the set \(\mathbb{R}^{n} \setminus \bigcup\{a_{i} + L_{i}\}\), where the union is taken over all indices \(i\) such that \(\mathrm{tail}(U_{i})\) is a vector space.
Assume that all the linear spaces \(V_{i} = \mathrm{tail}(U_{i})\) above are equal to the same linear space \(V\) and denote by \(T\) a subspace transversal to \(V\), i.e., a linear subspace of \(\mathbb{R}^{n}\) such that \(\mathbb{R}^{n} = T \oplus V\). Then the set \(\mathbb{R}^{N} \setminus U\) is homotopy equivalent to \(T \setminus \{b_{i}\}\), where \(b_{i}:= T \cap \{a_{i} + V_{i}\}\). The statement describes completely the homotopy type of the set \(\mathbb{R}^{n} \setminus \bigcup H_{i}\), where \(\{H_{i}\}\) is any collection of affine hyperplanes in \(\mathbb{R}^{n}\), and this comes from the fact that the complement \(\mathbb{R}^{n}\setminus \bigcup H_{i}\) is a union of open convex sets.
The second part of the paper is devoted to volumes of generalized virtual polytopes, and this is done to study the cohomology rings of generalized quasitoric manifolds.
Suppose that \(\triangle\) is a triangulation of an \((n-1)\)-dimensional sphere on the vertex set \(V(\triangle) = \{v_{1}, \dots, v_{m}\}\). In what follows, we will identify a simplex of \(\triangle\) with the set of its vertices viewed as a subset in \(\{1, \dots, m\}\).
A map \(\lambda : V(\triangle) \rightarrow (\mathbb{R}^{n})^{*}\) is called a characteristic map if for any vertices \(v_{i_{1}}, \dots, v_{i_{r}}\) belonging to the same simplex of \(\triangle\) the images \(\lambda(v_{i_{1}}), \dots, \lambda(v_{i_{r}})\) are linearly independent over \(\mathbb{R}\). Such a map defines an \(m\)-dimensional family of hyperplane arrangements \(\mathcal{AP}\) in the following way. For any \(h = (h_{1}, \dots, h_{m}) \in \mathbb{R}^{m}\), the arrangement \(\mathcal{AP}\) is given by
\[
\mathcal{AP} = \{H_{1}, \dots, H_{m}\} \quad \text{ with } \quad H_{i} = \{\ell_{i}(x) = h_{i}\},
\]
where \(\ell_{i}\) denotes the linear function \(\lambda(v_{i})\) for each \(i \in \{1, \dots, m\}\). Given a subset \(I \subset \{1, \dots,m\}\), we also define \(H_I = \bigcap_{j \in I} H_{j}\). If \(I \in \triangle\), then \(\Gamma_{I}\) denotes the face dual to \(I\) in the polyhedral complex \(\triangle^{\perp}\) dual to the simplicial complex \(\triangle\). A generalized virtual polytope is defined as a map
\[
f \, : \triangle^{\perp} \rightarrow \bigcup_{\mathcal{AP}(h)}H_{i}
\]
subordinate to the characteristic map \(\lambda\), that is for any \(I \subset \{1, \dots, m\}\) one has
\[
f(\Gamma_{I}) \subset H_{I}.
\]
Let \(U\) be a bounded region of \(\mathbb{R}^{n} \setminus \bigcup_{\mathcal{AP}(h)}H_{i}\) and \(W(U,f)\) be a winding number of \(f\). For a given polynomial \(Q\) on \(\mathbb{R}^{n}\), consider
\[
I_{Q}(f) := \sum W(U,f) \int_{U}Q\omega .
\]
Theorem F. Let \(I \subset \{i_{1}, \dots, i_{r}\} \subset \{1, \dots, m\}\) be such that \(I \not\in \triangle\) and \(k_{1}, \dots, k_{r}\) be positive integers. Then we have
\[
\partial_{i_{1}}^{k_{1}} \cdots \partial_{i_{r}}^{k_{r}}(I_{Q})(f) = 0.
\]
However, if \(r=n=\dim \triangle +1\) and \(I\) is a simplex in \(\triangle\) dual to a vertex \(A \in \triangle^{\perp}\), then we have
\[
\partial_{I}(I_{Q})(f) = \mathrm{sgn}(I)Q(A)|\det(e_{i_{1}}, \dots, e_{i_{n}})|.
\]
In the last part of the paper, the authors describe the cohomology rings of quasitoric manifolds. Assume that \(K = K_{\Sigma}\) is a star-shaped sphere which is an intersection of a complete simplicial fan \(\Sigma\) in \(\mathbb{R}^{n} \simeq N \otimes_{\mathbb{Z}}\mathbb{R}\) with the unit sphere \(S^{n-1} \subset \mathbb{R}^{n}\). Let \(\mathcal{Z}_{K}\) be the moment-angle complex and denote by \(\Lambda : \Sigma(1) \rightarrow N\) a characteristic map. Then the \((m-n)\)-dimensional subtorus \(H_{\Lambda} := \ker \exp \lambda \subset (S^{1})^{m}\) acts freely on \(\mathcal{Z}_{K}\), and the smooth manifold \(X_{\Sigma,\Lambda} :=\mathcal{Z}_{K} / H_{\Lambda}\) is called a generalized quasitoric manifold. In that setting, the authors show the following results.
Theorem G. Let \(X_{\Sigma, \Lambda}\) be a generalized quasitoric manifold. Then \(X_{\Sigma, \Lambda}\) has a cellular decomposition with only even-dimensional cells. The cells in this decomposition are in a bijection with the maximal cones \(\tau\) in \(\Sigma\), and the Euler characteristic of the manifold \(X_{\Sigma,\Lambda}\) is equal to the number of maximal cones in \(\Sigma\).
As it is pointed out by the authors, in order to describe the cohomology ring of a generalized quasitoric manifold, it is enough to compute the self-intersection polynomial
\[
h_{1}[D_{1}] + \dots + h_{m}[D_{m}] \mapsto \langle(h_{1}[D_{1}] + \dots + h_{m}[D_{m}])^{m}, [X_{\Sigma,\Lambda}]\rangle
\]
on the space of all linear combinations of classes of codimension \(2\) characteristic submanifolds.
Theorem H. Let \(X_{\Sigma,\Lambda}\) be a generalized quasitoric manifold with codimension \(2\) characteristic submanifolds \(D_{1}, \dots, D_{m}\). Then the following identity holds:
\[
\langle(h_{1}[D_{1}] + \dots + h_{m}[D_{m}])^{m}, [X_{\Sigma,\Lambda}]\rangle = n! \cdot \mathrm{Vol}(f_{h}),
\]
where \(f_{h} \in \mathcal{P}_{\Sigma,\Lambda}\) is a generalized virtual polytope associated with the simplicial complex \(K_{\Sigma}\), the characteristic map \(\Lambda\), and the set of parameters \(h = (h_{1}, \dots, h_{m})\).
Theorem I. Let \(X_{\Sigma, \Lambda}\) be a generalized quasitoric manifold and let \(\mathcal{P}_{\Sigma,\Lambda}\) be the space of generalized virtual polytopes associated with it. Then the cohomology ring \(H^{\ast}(X_{\Sigma,\Lambda})\) can be computed as
\[
H^{\ast}(X_{\Sigma, \Lambda}) = \mathrm{Diff}(\mathcal{P}_{\Sigma, \Lambda})/ \mathrm{Ann}(\mathrm{Vol}),
\]
where \(\mathrm{Diff}(\mathcal{P}_{\Sigma,\Lambda})\) is the ring of differential operators with constant coefficients on \(\mathcal{P}_{\Sigma,\Lambda}\) and \(\mathrm{Ann}(\mathrm{Vol})\) is the annihilator ideal of the volume polynomial.
Reviewer: Piotr Pokora (Kraków)A lower bound for the area of plateau foamshttps://zbmath.org/1503.531202023-03-23T18:28:47.107421Z"Gimeno, Vicent"https://zbmath.org/authors/?q=ai:gimeno.vicent"Markvorsen, Steen"https://zbmath.org/authors/?q=ai:markvorsen.steen"Sotoca, José M."https://zbmath.org/authors/?q=ai:sotoca.jose-mSummary: Real foams can be viewed as geometrically well-organized dispersions of more or less spherical bubbles in a liquid. When the foam is so drained that the liquid content significantly decreases, the bubbles become polyhedral-like and the foam can be viewed now as a network of thin liquid films intersecting each other at the Plateau borders according to the celebrated Plateau's laws. In this paper we estimate from below the surface area of a spherically bounded piece of a foam. Our main tool is a new version of the divergence theorem which is adapted to the specific geometry of a foam with special attention to its classical Plateau singularities. As a benchmark application of our results, we obtain lower bounds for the fundamental cell of a Kelvin foam, lower bounds for the so-called cost function, and for the difference of the pressures appearing in minimal periodic foams. Moreover, we provide an algorithm whose input is a set of isolated points in space and whose output is the best lower bound estimate for the area of a foam that contains the given set as its vertex set.Classification of angular curvature measures and a proof of the angularity conjecturehttps://zbmath.org/1503.531412023-03-23T18:28:47.107421Z"Wannerer, Thomas"https://zbmath.org/authors/?q=ai:wannerer.thomas\textit{A. Bernig} et al. [Geom. Funct. Anal. 24, No. 2, 403--492 (2014; Zbl 1298.53074)] stated the so-called angularity conjecture, which reads as follows: the space of angular curvature measures on a Riemannian manifold is invariant under the action of the Lipschitz-Killing algebra. The main goal of the present paper is to confirm the conjecture. Besides, a complete classification of translation-invariant angular curvature measures on \(\mathbb{R}^{n}\) is obtained.
Valuation theory on convex spaces was extended to Riemannian manifolds by means of Lipschitz-Killing valuations.
\textit{S. Alesker} in a series of papers published from 2001 to 2010 [Geom. Funct. Anal. 11, No. 2, 244--272 (2001; Zbl 0995.52001); J. Differ. Geom. 63, No. 1, 63--95 (2003; Zbl 1073.52004); Isr. J. Math. 156, 311--339 (2006; Zbl 1132.52017); Adv. Math. 207, No. 1, 420--454 (2006; Zbl 1117.52016); Geom. Funct. Anal. 17, No. 4, 1321--1341 (2007; Zbl 1132.52018); Lect. Notes Math. 1910, 1--44 (2007; Zbl 1127.52016); Geom. Funct. Anal. 20, No. 5, 1073--1143 (2010; Zbl 1213.52013)] stated the existence of such theory in the Riemannian context.
The paper under review is well motivated. The author needs to use technical results of different branches of mathematics, such as convex and Riemannian geometries and representation theory, and some relevant results, such as the Nash isometric embedding theorem. All of this is carefully explained throughout the paper.
Reviewer: Fernando Etayo Gordejuela (Santander)2-systems of arcs on spheres with prescribed endpointshttps://zbmath.org/1503.570132023-03-23T18:28:47.107421Z"Douba, Sami"https://zbmath.org/authors/?q=ai:douba.samiLet \(S\) be an \(n\)-punctured sphere with \(n \geq 3\). A \(k\)-system of arcs on \(S\) is a collection \(\mathcal{A}\) of essential simple arcs on \(S\) such that if \(\alpha, \beta \in \mathcal{A}\) then (i) \(\alpha\) is not homotopic to \(\beta\) and (ii) the geometric intersection number of \(\alpha\) and \(\beta\) is at most \(k\).
\textit{P. Przytycki} [Geom. Funct. Anal. 25, No. 2, 658--670 (2015; Zbl 1319.57016)] proved that the maximum size of a 1-system of arcs on \(S\) joining two prescribed (not necessarily distinct) punctures \(p\) and \(q\) of \(S\) is \(\binom{n-1}{2}\). \textit{A. Bar-Natan} [Groups Geom. Dyn. 14, No. 4, 1309--1332 (2020; Zbl 1470.57029)] showed that the maximum size of a 2-system of arcs on \(S\) starting and ending at a prescribed puncture \(p\) is \(\binom{n}{3}\). Theorem 1.3 in this article shows that Bar-Natan's result holds even when the arcs start and end at two distinct fixed punctures \(p\) and \(q\) of \(S\). More precisely, it shows that the maximum size of a 2-system of arcs on \(S\) joining two prescribed distinct punctures \(p\) and \(q\) of \(S\) is \(\binom{n}{3}\). So together, Bar-Natan's result and Theorem 1.3 of the current article, show that the maximum size of a 2-system of arcs on \(S\) joining two prescribed (not necessarily distinct) punctures \(p\) and \(q\) of \(S\) is \(\binom{n}{3}\).
The author gives an explicit example of a 2-system of arcs joining a fixed pair of distinct punctures on \(S\) whose size is \(\binom{n}{3}\) hence proving that the bound in Theorem 1.3 is tight.
The proof of Theorem 1.3 is by induction on the number of punctures of \(S\), the base case being \(n=3\). A key Lemma 1.4 in the article is used to control the number of arcs that become homotopic after forgetting a puncture on \(S\). To prove Lemma 1.4, the author uses annular square diagrams and in doing so, proves Theorem 1.5 that asserts that a 1-system annular diagram is either a cycle or has a corner on each of its boundary paths. A direct proof of a result given by the referee of the article which circumvents the need to venture into annular diagrams for the proof of Lemma 1.4 is also included.
Reviewer: Sreekrishna Palaparthi (Guwahati)A note on geometric duality in matroid theory and knot theoryhttps://zbmath.org/1503.570222023-03-23T18:28:47.107421Z"Traldi, Lorenzo"https://zbmath.org/authors/?q=ai:traldi.lorenzoThe author establishes in this paper the observation that for planar graphs, the geometric duality relation generates both 2-isomorphism and abstract duality. ``This observation has the surprising consequence that for links, the equivalence relation defined by isomorphisms of checkerboard graphs is the same as the equivalence relation defined by 2-isomorphisms of checkerboard graphs.'' Despite this fact, the author proves that geometric duality suffices to define abstract duality of planar graphs. Moreover, he shows that geometric duality also suffices to define 2-isomorphism of planar graphs.
Reviewer: Ismail Naci Cangül (Bursa)Existence of continuous maps from \(d\)-spheres \((d \geq 1)\) to its various triangulations having the disjoint support propertyhttps://zbmath.org/1503.570272023-03-23T18:28:47.107421Z"Choudhury, Snigdha Bharati"https://zbmath.org/authors/?q=ai:choudhury.snigdha-bharati"Deo, Satya"https://zbmath.org/authors/?q=ai:deo.satya.1"Podder, Shubhankar"https://zbmath.org/authors/?q=ai:podder.shubhankarThe disjoint support property in the title means that a continuous map \(h\) from the \(d\)-sphere \(S^d\) to a triangulated \(d\)-sphere \(\Sigma^d\) satisfies: For any two antipodal points \(x,-x\in S^d\) the images \(h(x)\), \(h(-x)\) are contained in two disjoint faces of the triangulation. For \(\Sigma^d=\partial\Delta^{d+1}\) such maps exist but no homeomorphism. The central projection from \(S^d\) onto the boundary of the \((d+1)\)-dimensional cross polytope = the convex hull of \((1,0,\dots,0),\dots,(0,0,\dots,1)\in \mathbb{R}^{d+1}\) provides an example of such a homeomorphism. In general, it seems that the central projection from \(S^d\) onto the boundary of any centrally symmetric \((d+1)\)-polytope (simplicial or not) has the disjoint support property.
On the other hand for any \(d\) the authors construct a triangulated \(\Sigma^d\) such that there is such a map \(h\) but no homeomorphism. These examples are based on iterated suspensions of certain stacked \(3\)-polytopes presented in a previous article by the first two authors [Period. Math. Hung. 83, No. 1, 20--31 (2021; Zbl 1488.52017)].
Reviewer: Wolfgang Kühnel (Stuttgart)The \(\beta\)-Delaunay tessellation: description of the model and geometry of typical cellshttps://zbmath.org/1503.600142023-03-23T18:28:47.107421Z"Gusakova, Anna"https://zbmath.org/authors/?q=ai:gusakova.anna"Kabluchko, Zakhar"https://zbmath.org/authors/?q=ai:kabluchko.zakhar-a"Thäle, Christoph"https://zbmath.org/authors/?q=ai:thale.christophSummary: In this paper we introduce two new classes of stationary random simplicial tessellations, the so-called \(\beta\)- and \(\beta^{\prime}\)-Delaunay tessellations. Their construction is based on a space-time paraboloid hull process and generalizes that of the classical Poisson-Delaunay tessellation. We explicitly identify the distribution of volume-power-weighted typical cells, establishing thereby a remarkable connection to the classes of \(\beta\)- and \(\beta^{\prime}\)-polytopes. These representations are used to determine the principal characteristics of such cells, including volume moments, expected angle sums, and cell intensities.Improved polytope volume calculations based on Hamiltonian Monte Carlo with boundary reflections and sweet arithmeticshttps://zbmath.org/1503.650412023-03-23T18:28:47.107421Z"Chevallier, Augustin"https://zbmath.org/authors/?q=ai:chevallier.augustin"Pion, Sylvain"https://zbmath.org/authors/?q=ai:pion.sylvain"Cazals, Frédéric"https://zbmath.org/authors/?q=ai:cazals.fredericSummary: Computing the volume of a high dimensional polytope is a fundamental problem in geometry, also connected to the calculation of densities of states in statistical physics, and a central building block of such algorithms is the method used to sample a target probability distribution. This paper studies Hamiltonian Monte Carlo (HMC) with reflections on the boundary of domain, providing an enhanced alternative to Hit-and-run (HAR) to sample a target distribution restricted to the polytope. We make three contributions. First, we provide a convergence bound, paving the way to more precise mixing time analysis.\par Second, we present a robust implementation based on multi-precision arithmetic, a mandatory ingredient to guarantee exact predicates and robust constructions. We however allow controlled failures to happen, introducing the Sweeten Exact Geometric Computing (SEGC) paradigm. Third, we use our HMC random walk to perform H-polytope volume calculations, using it as an alternative to HAR within the volume algorithm by Cousins and Vempala. The systematic tests conducted up to dimension \(n=100\) on the cube, the isotropic and the standard simplex show that HMC significantly outperforms HAR both in terms of accuracy and running time. Additional tests show that calculations may be handled up to dimension \(n=500\). These tests also establish that multiprecision is mandatory to avoid exits from the polytope.The painter's problem: covering a grid with colored connected polygonshttps://zbmath.org/1503.682832023-03-23T18:28:47.107421Z"van Goethem, Arthur"https://zbmath.org/authors/?q=ai:van-goethem.arthur"Kostitsyna, Irina"https://zbmath.org/authors/?q=ai:kostitsyna.irina"van Kreveld, Marc"https://zbmath.org/authors/?q=ai:van-kreveld.marc-j"Meulemans, Wouter"https://zbmath.org/authors/?q=ai:meulemans.wouter"Sondag, Max"https://zbmath.org/authors/?q=ai:sondag.max"Wulms, Jules"https://zbmath.org/authors/?q=ai:wulms.jules-j-h-mSummary: Motivated by a new way of visualizing hypergraphs, we study the following problem. Consider a rectangular grid and a set of colors \(\chi\). Each cell \(s\) in the grid is assigned a subset of colors \(\chi_s\subseteq\chi\) and should be partitioned such that for each color \(c\in\chi_s\) at least one piece in the cell is identified with \(c\). Cells assigned the empty color set remain white. We focus on the case where \(\chi=\{\mathrm{red},\mathrm{blue}\}\). Is it possible to partition each cell in the grid such that the unions of the resulting red and blue pieces form two connected polygons? We analyze the combinatorial properties and derive a necessary and sufficient condition for such a painting. We show that if a painting exists, there exists a painting with bounded complexity per cell. This painting has at most five colored pieces per cell if the grid contains white cells, and at most two colored pieces per cell if it does not.
For the entire collection see [Zbl 1381.68007].On \(L^2\) convergence of the Hamiltonian Monte Carlohttps://zbmath.org/1503.810262023-03-23T18:28:47.107421Z"Ghosh, Soumyadip"https://zbmath.org/authors/?q=ai:ghosh.soumyadip"Lu, Yingdong"https://zbmath.org/authors/?q=ai:lu.yingdong"Nowicki, Tomasz"https://zbmath.org/authors/?q=ai:nowicki.tomasz-j|nowicki.tomaszSummary: We represent the abstract Hamiltonian (Hybrid) Monte Carlo (HMC) algorithm as iterations of an operator on densities in a Hilbert space, and recognize two invariant properties of Hamiltonian motion sufficient for convergence. Under a mild coverage assumption, we present a proof of strong convergence of the algorithm to the target density. The proof relies on the self-adjointness of the operator, and we extend the result to the general case of the motions beyond Hamiltonian ones acting on a finite dimensional space, to the motions acting an abstract space equipped with a reference measure, as long as they satisfy the two sufficient properties. For standard Hamiltonian motion, the convergence is also geometric in the case when the target density satisfies a log-convexity condition.Lattice-free simplices with lattice width \(2d - o(d)\)https://zbmath.org/1503.900772023-03-23T18:28:47.107421Z"Mayrhofer, Lukas"https://zbmath.org/authors/?q=ai:mayrhofer.lukas"Schade, Jamico"https://zbmath.org/authors/?q=ai:schade.jamico"Weltge, Stefan"https://zbmath.org/authors/?q=ai:weltge.stefanThis paper considers the properties of lattice-free convex bodies which are those that do not contain any integer points in their interior. Such bodies are flat with respect to the integer lattice and therefore the lattice width is an interesting property of lattice-free bodies. The authors propose and prove a new lower bound for the lattice width relating to a family of simplicities and present a number of useful lemmas. A suggested method for the construction of such simplicities is also presented. The article concludes with a list of open questions relating to the properties of lattice-free convex bodies.
For the entire collection see [Zbl 1492.90008].
Reviewer: Efstratios Rappos (Aubonne)