Recent zbMATH articles in MSC 52https://zbmath.org/atom/cc/522024-09-13T18:40:28.020319ZWerkzeugConvexity, squeezing, and the Elekes-Szabó theoremhttps://zbmath.org/1540.050092024-09-13T18:40:28.020319Z"Roche-Newton, Oliver"https://zbmath.org/authors/?q=ai:roche-newton.oliver"Wong, Elaine"https://zbmath.org/authors/?q=ai:wong.elaineSummary: This paper explores the relationship between convexity and sum sets. In particular, we show that elementary number theoretical methods, principally the application of a squeezing principle, can be augmented with the Elekes-Szabó Theorem in order to give new information. Namely, if we let \(A \subset \mathbb{R}\), we prove that there exist \(a, a^\prime \in A\) such that \[\left | \frac{(aA+1)^{(2)}(a^\prime A+1)^{(2)}}{(aA+1)^{(2)}(a^\prime A+1)} \right | \gtrsim |A|^{31/12}.\] We are also able to prove that \[\max \{|A+ A-A|, |A^2+A^2-A^2|, |A^3 + A^3 - A^3|\} \gtrsim |A|^{19/12}.\] Both of these bounds are improvements of recent results and takes advantage of computer algebra to tackle some of the computations.Matroid products in tropical geometryhttps://zbmath.org/1540.050272024-09-13T18:40:28.020319Z"Anderson, Nicholas"https://zbmath.org/authors/?q=ai:anderson.nicholas|anderson.nicholas-b.1Summary: Symmetric powers of matroids were first introduced by \textit{L. Lovász} [in: Comb. Surv., Proc. 6th Br. Comb. Conf., 45--86 (1977; Zbl 0361.05027)] and \textit{J. H. Mason} [in: Algebraic methods in graph theory, Vol. II, Conf. Szeged 1978, Colloq. Math. Soc. János Bolyai 25, 519--561 (1981; Zbl 0477.05022)] in the 1970s, where it was shown that not all matroids admit higher symmetric powers. Since these initial findings, the study of matroid symmetric powers has remained largely unexplored. In this paper, we establish an equivalence between valuated matroids with arbitrarily large symmetric powers and tropical linear spaces that appear as the variety of a tropical ideal. In establishing this equivalence, we additionally show that all tropical linear spaces are connected through codimension one. These results provide additional geometric and algebraic connections to the study of matroid symmetric powers, which we leverage to prove that the class of matroids with second symmetric power is minor-closed and has infinitely many forbidden minors.Small separations in pinch-graphic matroidshttps://zbmath.org/1540.050302024-09-13T18:40:28.020319Z"Guenin, Bertrand"https://zbmath.org/authors/?q=ai:guenin.bertrand"Heo, Cheolwon"https://zbmath.org/authors/?q=ai:heo.cheolwonSummary: Even-cycle matroids are elementary lifts of graphic matroids. Pinch-graphic matroids are even-cycle matroids that are also elementary projections of graphic matroids. In this paper we analyze the structure of 1-, 2-, and 3-separations in these matroids. As a corollary we obtain a polynomial-time algorithm that reduces the problem of recognizing pinch-graphic matroids to internally 4-connected matroids. Combining this with earlier results [the authors, Lect. Notes Comput. Sci. 12125, 182--195 (2020; Zbl 1504.05052); Math. Program. 204, No. 1--2 (A), 113--134 (2024; Zbl 1533.05046)] we obtain a polynomial-time algorithm for recognizing even-cycle matroids and we obtain a polynomial-time algorithm for recognizing even-cut matroids.Maximum area converses of Harary-Harborth polyform theoremshttps://zbmath.org/1540.050332024-09-13T18:40:28.020319Z"Yang, Winston C."https://zbmath.org/authors/?q=ai:yang.winston-c"Meyer, Robert R."https://zbmath.org/authors/?q=ai:meyer.robert-rSummary: \textit{F. Harary} and \textit{H. Harborth} [J. Comb. Inf. Syst. Sci. 1, 1--8 (1976; Zbl 0402.05055)] minimized the number of arcs in a polyform with a fixed number of cells. They also gave a ``swirl algorithm'' for constructing such polyforms. \par Using constrained optimization, we give ``converse'' (inverse) results; we find the max number of cells of a polyform with a fixed number of exterior arcs and show how the swirl algorithm constructs such polyforms. \par Using exterior-arc versions of Harary and Harborth's min-arc formulas [loc. cit.], we show how their formulas and swirl algorithms are consequences of the max-cell results.Flexibility and rigidity of frameworks consisting of triangles and parallelogramshttps://zbmath.org/1540.050432024-09-13T18:40:28.020319Z"Grasegger, Georg"https://zbmath.org/authors/?q=ai:grasegger.georg"Legerský, Jan"https://zbmath.org/authors/?q=ai:legersky.janSummary: A framework, which is a (possibly infinite) graph with a realization of its vertices in the plane, is called flexible if it can be continuously deformed while preserving the edge lengths. We focus on flexibility of frameworks in which 4-cycles form parallelograms. For the class of frameworks considered in this paper (allowing triangles), we prove that the following are equivalent: flexibility, infinitesimal flexibility, the existence of at least two classes of an equivalence relation based on 3- and 4-cycles and being a non-trivial subgraph of the Cartesian product of graphs. We study the algorithmic aspects and the rotationally symmetric version of the problem. The results are illustrated on frameworks obtained from tessellations by regular polygons.A geometric proof for the root-independence of the greedoid polynomial of Eulerian branching greedoidshttps://zbmath.org/1540.050682024-09-13T18:40:28.020319Z"Tóthmérész, Lilla"https://zbmath.org/authors/?q=ai:tothmeresz.lillaSummary: We define the root polytope of a regular oriented matroid, and show that the greedoid polynomial of an Eulerian branching greedoid rooted at vertex \(v_0\) is equivalent to the \(h^{\ast}\)-polynomial of the root polytope of the dual of the graphic matroid.
As the definition of the root polytope is independent of the vertex \(v_0\), this gives a geometric proof for the root-independence of the greedoid polynomial for Eulerian branching greedoids, a fact which was first proved by \textit{S. H. Chan} [ibid. 154, 145--171 (2018; Zbl 1373.05075)], \textit{K. Perrot} and \textit{T. van Pham} [Electron. J. Comb. 23, No. 1, Research Paper P1.57, 34 p. (2016; Zbl 1335.91023)] using sandpile models. We also obtain that the greedoid polynomial does not change if we reverse every edge of an Eulerian digraph.On book crossing numbers of the complete graphhttps://zbmath.org/1540.051252024-09-13T18:40:28.020319Z"Abrego, Bernardo"https://zbmath.org/authors/?q=ai:abrego.bernardo-m"Kinzel, Julia"https://zbmath.org/authors/?q=ai:kinzel.julia"Fernandez-Merchant, Silvia"https://zbmath.org/authors/?q=ai:fernandez-merchant.silvia"Lagoda, Evgeniya"https://zbmath.org/authors/?q=ai:lagoda.evgeniya"Sapozhnikov, Yakov"https://zbmath.org/authors/?q=ai:sapozhnikov.yakovSummary: A \(k\)-page book drawing of a graph \(G\) is a drawing of \(G\) on \(k\) halfplanes with a line \(l\) as a common boundary such that the vertices are located on \(l\) and the edges cannot cross \(l\). The \(k\)-page book crossing number of the graph \(G\), denoted by \(\nu_k(G)\), is the minimum number of edge-crossings over all \(k\)-page book drawings of \(G\). This paper improves previous results on \(k\)-page book crossing numbers of the complete graph \(K_n\). We determine \(\nu_k(K_n)\) whenever \(2<n/k\leq 3\) and improve the lower bounds on \(\nu_k(K_n)\) for all \(k\geq 14\). Our proofs rely on bounding the number of edges in convex geometric graphs with few crossings per edge.Bounds on the defect of an octahedron in a rational latticehttps://zbmath.org/1540.110872024-09-13T18:40:28.020319Z"Fadin, Mikhail"https://zbmath.org/authors/?q=ai:fadin.mikhailIn what follows let \(\Gamma\subset \mathbb{R}^n\) be an an n-dimensional lattice in Euclidean space containing the origin, with basis \(\xi=(e_1,e_2,\ldots,e_n)\). If \(\Gamma\) is a sub-lattice of the lattice \(\Lambda\), then \(\Lambda\) is referred to as a centering of the lattice \(\Gamma\).
Furthermore define the defect of the basis \(\xi\), with respect to the lattice \(\Lambda\), to be the smallest integer \(d\) such that \((n-d)\) vectors chosen from \(\xi\), together with \(d\) vectors chosen from the lattice \(\Lambda\) form a basis for \(\Lambda\). The defect of the basis is denoted by \(d(\xi, \Lambda)\) or simply \(d\).
For a given basis \(\xi\), an octahedron corresponding to \(\xi\) is defined as the set
\[
\mathcal{O}_\xi^n=\left \{ x\in \mathbb{R}^n: x=\lambda_1 e_1+\ldots +\lambda_n e_n; |\lambda_1|+\ldots +|\lambda_n|\leq 1\right \}.
\]
The octahedron \(\mathcal{O}_\xi^n\) is called admissible with respect to the lattice \(\Lambda\) if its interior contains no points of \(\Lambda\), except for the origin and \(\pm e_i,\,\,\, 1\leq i\leq n\).
For an octahedron \(\mathcal{O}_\xi^n\), corresponding to the basis \(\xi\), that is admissible with respect to the centering lattic \(\Lambda\), the quantity \(d(\xi, \Lambda)\) is denoted by \(d(\mathcal{O}_\xi^n, \Lambda)\) and is referred to as the defect of the admissible octahedron \(\mathcal{O}_\xi^n\) in the lattice \(\Lambda\).
In this present work the author considers the case \(\Gamma=\mathbb{Z}^n\), with \(\xi\) the standard unit basis in the directions of the coordinate axes. They are interested in the property
\[
d_n = \text{max}_{\Lambda \in \mathcal{A}_n} d(\mathcal{O}_\xi^n, \Lambda),
\]
where \(\mathcal{A}_n\) is the set of all rational lattices \(\Lambda\) such that \(\mathcal{O}_\xi^n\) is admissible with respect to \(\Lambda\). In conjunction with the result of Konyagin's, which says that there exists a positive constant \(C\) such that \(n-d_n \leq C \log n\), the author shows that
\[
n- d_n = O(\log n).
\]
Further bounds on generalisations of \(d_n\) are also established.
Reviewer: Matthew C. Lettington (Cardiff)Minimal lattice points in the Newton polyhedron and application to normal idealshttps://zbmath.org/1540.130132024-09-13T18:40:28.020319Z"Al-Ayyoub, Ibrahim"https://zbmath.org/authors/?q=ai:al-ayyoub.ibrahimLet \(S\) be a ring and \(I\) be an ideal in \(S\). An element \(f\in S\) is \textit{integral} over \(I\), if there exists an equation
\[
f^k+c_1f^{k-1}+\cdots +c_{k-1}f+c_k=0\text{ with }c_i\in I^i.
\]
The set of elements \(\overline{I}\) in \(S\) which are integral over \(I\) is the \textit{integral closure} of \(I\). The ideal \(I\) is called \textit{integrally closed}, if \(I=\overline{I}\), and \(I\) is said to be \textit{normal} if all powers of \(I\) are integrally closed. Now, consider a monomial ideal \(I=(x_1^{a_1}, \ldots, x_n^{a_n}) \subset R=K[x_1, \ldots, x_n]\) with each \(a_i\) is a positive integer and \(K\) is a field. Let \(\mathbf{I}(a_1, \ldots, a_n)\) denote the integral closure of the ideal \(I\). The main aim of this paper is to present an elementary and simpler proof of Theorem 5.1 in [\textit{L. Reid} et al., Commun. Algebra 31, No. 9, 4485--4506 (2003; Zbl 1021.13008)]. To do this, the author uses the elementary definition of convex sets, and in particular, a simple characterization of the exponents of the minimal generators of \(\mathbf{I}(a_1, \ldots, a_n)\). In fact, let \(L= \mathbf{I}(a_1, \ldots, a_n, a_{n+1} +l)\) and \(J=\mathbf{I}(a_1, \ldots, a_n, a_{n+1})\), where \(l=\mathrm{lcm}(a_1, \ldots, a_n)\). Then the author proves that:
{Corollary 1.} If \(L\) is normal, then \(J\) is so.
{Corollary 2.} If \(J\) is normal and \(a_{n+1} \geq l\), then \(L\) is so.
Reviewer: Mehrdad Nasernejad (Lens)Hilbert-Poincaré series of matroid Chow rings and intersection cohomologyhttps://zbmath.org/1540.140092024-09-13T18:40:28.020319Z"Ferroni, Luis"https://zbmath.org/authors/?q=ai:ferroni.luis"Matherne, Jacob P."https://zbmath.org/authors/?q=ai:matherne.jacob-p"Stevens, Matthew"https://zbmath.org/authors/?q=ai:stevens.matthew"Vecchi, Lorenzo"https://zbmath.org/authors/?q=ai:vecchi.lorenzoThe authors study the Hilbert series of four objects arising in the Chow-theoretic and Kazhdan-Lusztig framework of matroids. These are, respectively, the Hilbert series of the Chow ring, the augmented Chow ring, the intersection cohomology module, and its stalk at the empty flat. The authors develop an explicit parallelism between the Kazhdan-Lusztig polynomial of a matroid and the Hilbert-Poincaré series of its Chow ring. This extends to a parallelism between the Z-polynomial of a matroid and the Hilbert-Poincaré series of its augmented Chow ring. This suggests to bring ideas from one framework to the other. Two main motivations are the real-rootedness conjecture for all of these polynomials, and the problem of computing them.
Uniform matroids are a case of combinatorial interest, where the authors link the resulting polynomials with certain real-rooted families appearing in combinatorics such as the Eulerian and the binomial Eulerian polynomials, and thus, they prove the real-rootedness of the Hilbert series of the augmented Chow rings of uniform matroids. In addition, they prove a version of a conjecture of Gedeon in the Chow setting: uniform matroids maximize coefficient-wise these polynomials for matroids with fixed rank and cardinality. By relying on the nonnegativity of the coefficients of the Kazhdan-Lusztig polynomials and the semi-small decompositions of Braden, Huh, Matherne, Proudfoot, and Wang, the authors strengthen the unimodality of the Hilbert series of Chow rings, augmented Chow rings, and intersection cohomologies to \(\gamma\)-positivity.
Reviewer: Mee Seong Im (Annapolis)The infection of \(\mathbb{Z}^2\)https://zbmath.org/1540.141032024-09-13T18:40:28.020319Z"Altmann, Klaus"https://zbmath.org/authors/?q=ai:altmann.klausThe derived category \(\mathcal D(X)\) of coherent sheaves on an algebraic variety \(X\) is a very interesting homological invariant in algebraic geometry. To gain a better understanding of \(\mathcal D(X)\), one may look for exceptional objects, that is, \(E \in \mathcal D(X)\) such that \(\mathrm{RHom}(E,E)= \mathbb C\cdot \mathrm{id}\). This allows to decompose \(\mathcal D(X)\) into the triangulated category generated by \(E_1=E\), which is equivalent to the category of graded vector spaces, and the category \({}^\perp E\) consisting of all objects \(F \in \mathcal D(X)\) with \(\mathrm{RHom}(F,E)=0\). By iterating this process for \({}^\perp E\), one may hope to arrive at a full exceptional sequences \((E_1,\ldots,E_m)\), that is,
\begin{itemize}
\item all \(E_i\) are exceptional objects;
\item \(\mathrm{RHom}(E_j,E_i)=0\) for \(i<j\);
\item the triangulated category generated by all \(E_i\) coincides with \(\mathcal D(X)\).
\end{itemize}
In the preprint [``The structure of exceptional sequences on toric varieties of Picard rank two'', Preprint, \url{arXiv:2112.14637}] by \textit{K. Altmann} and \textit{F. Witt}, full exceptional sequences of line bundles are completely classified for smooth projective toric varieties of Picard rank \(2\). Whereas in [J. Algebr. Comb. 56, No. 2, 305--322 (2022; Zbl 07572731)] by \textit{K. Altmann} and \textit{M. Altmann}, full exceptional sequences of line bundles on \((\mathbb P^1)^3\) are studied. Both articles have in common, that they translate the homological properties from above into a discrete setup. Line bundles are exceptional objects, and Pic\((X)\) is either \(\mathbb Z^2\) or \(\mathbb Z^3\) in the above settings. The vanishing conditions can be formulated also easily into a condition on points in \(\mathbb Z^2\) or \(\mathbb Z^3\). The last condition on generation is the trickiest. Both articles use that it is sufficient to check whether an exceptional sequence generates all line bundles, and show how to obtain from certain configurations of line bundles in the sequence new ones outside, and by iterating the process all line bundles.
The article under review appeared in Elemente der Mathematik, which wants to provide survey articles about current research for a broad audience. For this purpose, the author reverses the approach of the articles cited above. First one tries to find some sets in \(\mathbb Z^2\) satisfying some strange relation (which captures the vanishing condition on line bundles), and then given some combinatorial rules how to produce new points, yielding the whole \(\mathbb Z^2\) in the end (which the author calls infecting the plane; which corresponds to generating). The whole procedure can be explained in very elementary terms.
The author starts (in disguise) with the easiest case of \(\mathbb P^1 \times \mathbb P^1\), treating then also the general case of a smooth projective variety of Picard rank \(2\), and ending with the three-dimensional case of \((\mathbb P^1)^3\). Only in the end of the article, the author gives the proper geometric context of the combinatorics (in contrast to the order of this review).
Reviewer: Andreas Hochenegger (Milano)Castelnuovo polytopeshttps://zbmath.org/1540.141072024-09-13T18:40:28.020319Z"Tsuchiya, Akiyoshi"https://zbmath.org/authors/?q=ai:tsuchiya.akiyoshiSummary: It is known that the sectional genus of a polarized variety has an upper bound, which is an extension of the Castelnuovo bound on the genus of a projective curve. Polarized varieties whose sectional genus achieve this bound are called Castelnuovo. On the other hand, a lattice polytope is called Castelnuovo if the associated polarized toric variety is Castelnuovo. Kawaguchi characterized Castelnuovo polytopes having interior lattice points in terms of their \(h^{\ast}\)-vectors. In this paper, as a generalization of this result, we present a characterization of all Castelnuovo polytopes. Finally, as an application of our characterization, we give a sufficient criterion for a lattice polytope to be IDP.On minimal positive basis for polyhedral cones in \(\mathbb{R}^n\)https://zbmath.org/1540.150032024-09-13T18:40:28.020319Z"Alves, M. B."https://zbmath.org/authors/?q=ai:alves.mariane-branco|alves.magno-branco"Gomes, J. B."https://zbmath.org/authors/?q=ai:gomes.j-b"Pedroso, K. M."https://zbmath.org/authors/?q=ai:pedroso.kennedy-martinsSummary: For polyhedral cones in the Euclidean space, we present its conic dimension, which is invariant under linear isomorphisms that is sensitive to the number of generators of this cone, and the related notion of conic basis. We may interpret these two notions as versions of the definitions of linear dimension and linear basis for linear subspaces in the setting of polyhedral cones. We establish a conic version of the rank-nullity theorem that, in this case, is an inequality involving the conic dimensions of both the cone and its image under a linear map. We use this conic rank-nullity inequality to establish both a decomposition and a union of conic basis, involving the lineality space of the cone. We introduce the signature of a polyhedral cone and establish some results on the injectivity of a linear map and the preservation of the signature of a polyhedral cone under linear maps. In particular, we show that a linear map that acts injectively on the linear span of a polyhedral cone preserves its signature.Short note on some geometric inequalities derived from matrix inequalitieshttps://zbmath.org/1540.150222024-09-13T18:40:28.020319Z"Lombardi, Nico"https://zbmath.org/authors/?q=ai:lombardi.nico"Saorín Gómez, Eugenia"https://zbmath.org/authors/?q=ai:gomez.eugenia-saorinThere are many interesting connections between matrix analysis and convex geometry. This paper is mainly focused on some new examples about this interconnection. The main results mainly concern inequalities for a particular family of convex bodies, known as \(L_2\) zonoids. Such geometric inequalities are here reproved on the basis of classical inequalities for matrices.
Reviewer: Minghua Lin (Xi'an)The higher Stasheff-Tamari orders in representation theoryhttps://zbmath.org/1540.160142024-09-13T18:40:28.020319Z"Williams, Nicholas J."https://zbmath.org/authors/?q=ai:williams.nicholas-jSummary: We show that the relationship discovered by Oppermann and Thomas between triangulations of cyclic polytopes and the higher Auslander algebras of type \(A\), denoted \(A^d_n\), is an incredibly rich one. The \textit{higher Stasheff-Tamari orders} are two orders on triangulations of cyclic polytopes, defined in the 1990s by \textit{M. M. Kapranov} and \textit{V. A. Voevodskij} [Cah. Topologie Géom. Différ. Catégoriques 32, No. 1, 11--27 (1991; Zbl 0748.18010)], and \textit{P. H. Edelman} and \textit{V. Reiner} [Mathematika 43, No. 1, 127--154 (1996; Zbl 0854.06003)], who conjectured them to be equivalent. We first show that these orders correspond in even dimensions to natural orders on tilting modules defined by \textit{C. Riedtmann} and \textit{A. Schofield} [Comment. Math. Helv. 66, No. 1, 70--78 (1991; Zbl 0790.16013)] and studied by \textit{D. Happel} and \textit{L. Unger} [Algebr. Represent. Theory 8, No. 2, 147--156 (2005; Zbl 1110.16011)]. This result allows us to show that triangulations of odd-dimensional cyclic polytopes are in bijection with equivalence classes of \(d\)-maximal green sequences of \(A^d_n\), which we introduce as a higher-dimensional generalisation of the original maximal green sequences of Keller. We further interpret the higher Stasheff-Tamari orders in odd dimensions, where they correspond to natural orders on equivalences classes of \(d\)-maximal green sequences. The conjecture that these two partial orders on equivalence classes of \(d\)-maximal green sequences are equal amounts to an oriented version of the ``no-gap'' conjecture of \textit{T. Brüstle} et al. [Int. Math. Res. Not. 2014, No. 16, 4547--4586 (2014; Zbl 1346.16009)]. A corollary of our results is that this conjecture holds for \(A_n\), and that here the set of equivalence classes of (1-)maximal green sequences is a lattice.
For the entire collection see [Zbl 1530.16002].On the ranks of string C-group representations for symplectic and orthogonal groupshttps://zbmath.org/1540.200142024-09-13T18:40:28.020319Z"Brooksbank, Peter A."https://zbmath.org/authors/?q=ai:brooksbank.peter-aSummary: We determine the ranks of string C-group representations of the groups \(\mathrm{PSp}(4,\mathbb{F}_q)\cong\Omega (5,\mathbb{F}_q)\), and comment on those of higher-dimensional symplectic and orthogonal groups.
For the entire collection see [Zbl 1467.52001].Flow by \(\sigma_k\) curvature to the Orlicz Christoffel-Minkowski problemhttps://zbmath.org/1540.352542024-09-13T18:40:28.020319Z"Yi, Caihong"https://zbmath.org/authors/?q=ai:yi.caihongSummary: In this paper, we consider the anisotropic curvature flow of smooth, origin-symmetric, uniformly convex hypersurfaces in \(\mathbb{R}^{n+1}\). The flow exists for all time and converges smoothly to a solution of the even Orlicz Christoffel-Minkowski problem. Our proof also gives an approach to the solution of the \(L_p\) Christoffel-Minkowski problem.Measurable tilings by abelian group actionshttps://zbmath.org/1540.370382024-09-13T18:40:28.020319Z"Grebík, Jan"https://zbmath.org/authors/?q=ai:grebik.jan"Greenfeld, Rachel"https://zbmath.org/authors/?q=ai:greenfeld.rachel"Rozhoň, Václav"https://zbmath.org/authors/?q=ai:rozhon.vaclav"Tao, Terence"https://zbmath.org/authors/?q=ai:tao.terence-cSummary: Let \(X\) be a measure space with a measure-preserving action \((g,x)\mapsto g\cdot x\) of an abelian group \(G\). We consider the problem of understanding the structure of measurable tilings \(F\odot A=X\) of \(X\) by a measurable tile \(A\subset X\) translated by a finite set \(F\subset G\) of shifts, thus the translates \(f\cdot A,f\in F\) partition \(X\) up to null sets. Adapting arguments from previous literature, we establish a ``dilation lemma'' that asserts, roughly speaking, that \(F\odot A=X\) implies \(F^r\odot A=X\) for a large family of integer dilations \(r\), and use this to establish a structure theorem for such tilings analogous to that established recently by the second and fourth authors. As applications of this theorem, we completely classify those random tilings of finitely generated abelian groups that are ``factors of iid'', and show that measurable tilings of a torus \(\mathbb{T}^d\) can always be continuously (in fact linearly) deformed into a tiling with rational shifts, with particularly strong results in the low-dimensional cases \(d=1,2\) (in particular resolving a conjecture of \textit{C. T. Conley} et al. [``Divisibility of Spheres with Measurable Pieces'', Preprint, \url{arXiv:2012.07567}] in the \(d=1\) case).Periodic orbits on obtuse edge tessellating polygonshttps://zbmath.org/1540.370462024-09-13T18:40:28.020319Z"Baer, Benjamin R."https://zbmath.org/authors/?q=ai:baer.benjamin-r"Gilani, Faheem"https://zbmath.org/authors/?q=ai:gilani.faheem"Han, Zhigang"https://zbmath.org/authors/?q=ai:han.zhigang"Umble, Ronald"https://zbmath.org/authors/?q=ai:umble.ronald-nSummary: A periodic orbit on a frictionless billiard table is a piecewise linear path of a billiard ball that begins and ends at the same point with the same angle of incidence. The period of a primitive periodic orbit is the number of times the ball strikes a side of the table as it traverses its trajectory exactly once. In this paper we find and classify the periodic orbits on a billiard table in the shape of a 120-isosceles triangle, a 60-rhombus, a 60-90-120-kite, and a 30-right triangle. In each case, we use the edge tessellation (also known as tiling) of the plane generated by the figure to unfold a periodic orbit into a straight line segment and to derive a formula for its period in terms of the initial angle and initial position.On continuous billiard and quasigeodesic flows characterizing alcoves and isosceles tetrahedrahttps://zbmath.org/1540.370482024-09-13T18:40:28.020319Z"Lange, Christian"https://zbmath.org/authors/?q=ai:lange.christianClassical billiard reflection is not defined at nonsmooth points of the billiard boundary. For some billiard domains, it is possible to extend continuously the billiard reflection to such points, and one says that such domains admit a continuous billiard evolution.
In this paper, a complete characterisation of such convex polyhedral domains is given, by proving that they are exactly fundamental domains of discrete affine reflection groups. It is also shown that another class of domains admitting a continuous billiard evolution is given by convex domains with the boundary of class \(C^{2,1}\) with a positive definite second fundamental form. Moreover, their billiard trajectories, whose initial directions converge to a tangent vector of the boundary, converge to the geodesic of the boundary.
It is proved that the boundary of a convex polyhedron admits a continuous quasigeodesic flow if and only if it is a Riemannian orbifold with respect to its intrinsic metric. In the three-dimensional space, the only such polyhedra are isosceles tetrahedra.
It is conjectured that if a convex domain admits a continuous billiard evolution, then all its tangent cones are orbifolds.
Reviewer: Milena Radnović (Sydney)The Brouwer fixed point theorem and periodic solutions of differential equationshttps://zbmath.org/1540.470782024-09-13T18:40:28.020319Z"Cid, José Ángel"https://zbmath.org/authors/?q=ai:cid.jose-angel"Mawhin, Jean"https://zbmath.org/authors/?q=ai:mawhin.jean-lThis short note contains the proof of the equivalence of the well-known Brouwer fixed point theorem with a generalization, due to \textit{G. B. Gustafson} and \textit{K. Schmitt} [Proc. Am. Math. Soc. 42, 161--166 (1974; Zbl 0273.34054)], of a result by Krasnosel'skii concerning the existence of \(T\)-periodic solutions of a \(T\)-periodic differential equation satisfying a suitable tangency condition on the boundary of a convex set.
Actually, in the first part of the paper an alternative proof of Gustafson-Schmitt theorem is provided. In fact, the authors avoid the classical approach by locally Lipschitzian approximations of \(f\) requested by the use of the Poincaré map. Instead, they are able to apply directly the Brouwer fixed point theorem using convex analysis and the so-called Stampacchia method to study a \(T\)-periodic boundary value problem. The Stampacchia algorithm consists of a step-by-step approximation based on the variation of constants formula.
As the authors claim, despite of the impressive list of equivalent reformulations of the Brouwer fixed point theorem in the literature, the equivalence between the Brouwer fixed point theorem and the existence of periodic solutions for some differential equation seems to have been unnoticed.
The paper also contains some interesting historical notes.
Reviewer: Alessandro Calamai (Ancona)Convex geometries representable by at most five circles on the planehttps://zbmath.org/1540.520012024-09-13T18:40:28.020319Z"Adaricheva, Kira"https://zbmath.org/authors/?q=ai:adaricheva.kira"Bolat, Madina"https://zbmath.org/authors/?q=ai:bolat.madina"Daisy, Evan"https://zbmath.org/authors/?q=ai:daisy.evan"Garg, Ayush"https://zbmath.org/authors/?q=ai:garg.ayush"Ma, Grace"https://zbmath.org/authors/?q=ai:ma.grace"Olson, Michelle"https://zbmath.org/authors/?q=ai:olson.michelle"Pai, Rohit"https://zbmath.org/authors/?q=ai:pai.rohit"Raanes, Catherine"https://zbmath.org/authors/?q=ai:raanes.catherine"Riedel, Sean"https://zbmath.org/authors/?q=ai:riedel.sean"Rogge, Joseph"https://zbmath.org/authors/?q=ai:rogge.joseph"Sarch, Raviv S."https://zbmath.org/authors/?q=ai:sarch.raviv-s"Thompson, James"https://zbmath.org/authors/?q=ai:thompson.james-a|thompson.james-m|thompson.james-l|thompson.james-r"Yepez-Lopez, Fernanda"https://zbmath.org/authors/?q=ai:yepez-lopez.fernanda"Zhou, Stephanie"https://zbmath.org/authors/?q=ai:zhou.stephanieSummary: A convex geometry is a closure system satisfying the antiexchange property. We document all convex geometries on 4- and 5-element base sets with respect to their representation by circles on the plane. All 34 nonisomorphic geometries on a 4-element set can be represented by circles, and of 672 known geometries on a 5-element set, we give representations for 623. Of the 49 remaining geometries on a 5-element set, one was already shown not to be representable due to the weak carousel property, as articulated by \textit{K. Adaricheva} and \textit{M. Bolat} [Discrete Math. 342, No. 3, 726--746 (2019; Zbl 1409.52001)]. We show that seven more of these convex geometries cannot be represented by circles on the plane, due to what we term the \textit{triangular implications property}.On a formula for all sets of constant width in 3Dhttps://zbmath.org/1540.520022024-09-13T18:40:28.020319Z"Kawohl, Bernd"https://zbmath.org/authors/?q=ai:kawohl.bernd"Sweers, Guido"https://zbmath.org/authors/?q=ai:sweers.guido-hThis article provides a new parametrization of three-dimensional convex bodies of constant width. Let \(\varphi,\theta\) be standard parameters on the unit sphere sphere \(S^2\). The authors start with a function \(a\) of \(\varphi\) and \(\theta\) with explicitly given properties. This function determines uniquely another function \(h\) and a number \(r_0(a)\). A certain orthonormal basis \((\mathbf{U}(\varphi,\theta),\mathbf{V}(\varphi,\theta), \mathbf{W}(\theta))\) is defined. Then, for \(r\ge r_0(a)\),
\[
\mathbf{X}(\varphi,\theta)=\mathbf{X}_0 +\int_0^\varphi(r-a(s,\theta))\mathbf{V}(s,\theta)ds + h(\varphi,\theta)\mathbf{W}(\theta)
\]
describes the boundary of a body of constant width \(2r\). The parametrization is regular if \(r>r_0(a)\). Every three-dimensional body of constant width can be parametrized in this way. The proofs make use of a parametrization of two-dimensional domains of constant width that the authors have constructed earlier [\textit{B. Kawohl} and \textit{G. Sweers}, Commun. Pure Appl. Anal. 18, No. 4, 2117--2131 (2019; Zbl 1419.52001)]. Moreover, it uses older results on the determination of a convex body by selected projections. Altogether, the arguments are rather technical, and it remains to hope that the effort will be justified by an application. There are beautiful computer-generated illustrations.
Reviewer: Rolf Schneider (Freiburg im Breisgau)Two-dimensional self-trapping structures in three-dimensional spacehttps://zbmath.org/1540.520032024-09-13T18:40:28.020319Z"Manturov, V. O."https://zbmath.org/authors/?q=ai:manturov.vassily-olegovich"Kanel-Belov, A. Ya."https://zbmath.org/authors/?q=ai:kanel-belov.alexei"Kim, S."https://zbmath.org/authors/?q=ai:kim.seongjeong"Nilov, F. K."https://zbmath.org/authors/?q=ai:nilov.f-kSummary: It is known that a finite set of convex figures on the plane with disjoint interiors has at least one outermost figure, i.e., one that can be continuously moved ``to infinity'' (outside a large circle containing the other figures), while the other figures are left stationary and their interiors are not crossed during the movement. It has been discovered that, in three-dimensional space, there exists a phenomenon of self-trapping structures. A self-trapping structure is a finite (or infinite) set of convex bodies with non-intersecting interiors, such that if all but one body are fixed, that body cannot be ``carried to infinity.'' Since ancient times, existing structures have been based on the consideration of layers made of cubes, tetrahedra, and octahedra, as well as their variations. In this work, we consider a fundamentally new phenomenon of two-dimensional self-trapping structures: a set of two-dimensional polygons in three-dimensional space, where each polygonal tile cannot be carried to infinity. Thin tiles are used to assemble self-trapping decahedra, from which second-order structures are then formed. In particular, a construction of a column composed of decahedra is presented, which is stable when we fix two outermost decahedra, rather than the entire boundary of the layer, as in previously investigated structures.Convex geometry and its connections to harmonic analysis, functional analysis and probability theoryhttps://zbmath.org/1540.520042024-09-13T18:40:28.020319Z"Ball, Keith"https://zbmath.org/authors/?q=ai:ball.keith-mThe author describes important connections of Convex Geometry to Harmonic Analysis, Functional Analysis and Probability Theory.
The author starts with a discussion of some fundamental results in Convex Geometry: Brunn-Minkowski inequality, John's theorem on the maximal ellipsoid contained in a convex body and the Blaschke-Santaló inequality for the volume product of a convex body and its polar. A very important tool, the Steiner symmetrization is described. Also the question of differentiability and twice differentiability of the boundary of a convex body is discussed.
The second section describes the connections to Harmonic Analysis that means the reverse isoperimetric inequality, the inequality of Brascamp-Lieb, the theorem of Brenier-McCann on optimal transport, and the relation of projections and surface area.
In the third section connections to Functional Analysis are explained.The author starts with what is the most important connection to the Geometry of Banach spaces, the theorem of Dvoretzky on intersections of a convex body with a hyperplane that are almost ellipsoidal.
Furthermore, the reverse inequality to the Blaschke-Santaló inequality is discussed.
In the fourth section the relation of Convex Geometry and Probability Theory is presented. Among other things the slicing problem is discussed: Do we find for any \(n\)-dimensional, convex body of volume \(1\) a hyperplane of dimension \(n-1\) such that its intersection with the convex body has a volume that is greater than some universal constant that does not depend on the dimension.
For the entire collection see [Zbl 1532.00038].
Reviewer: Carsten Schütt (Kiel)On the Musielak-Orlicz-Gauss image problemhttps://zbmath.org/1540.520052024-09-13T18:40:28.020319Z"Huang, Qingzhong"https://zbmath.org/authors/?q=ai:huang.qingzhong"Xing, Sudan"https://zbmath.org/authors/?q=ai:xing.sudan"Ye, Deping"https://zbmath.org/authors/?q=ai:ye.deping"Zhu, Baocheng"https://zbmath.org/authors/?q=ai:zhu.baochengSummary: In the present paper we initiate the study of the Musielak-Orlicz-Brunn-Minkowski theory for convex bodies. In particular, we develop the Musielak-Orlicz-Gauss image problem aiming to characterize the Musielak-Orlicz-Gauss image measure of convex bodies. For a convex body \(K\), its Musielak-Orlicz-Gauss image measure, denoted by \(\tilde{C}_{\Theta}(K,\cdot)\), involves a triple \(\Theta=(G,\Psi,\lambda)\) where \(G\) and \(\Psi\) are two Musielak-Orlicz functions defined on \(S^{n-1}\times(0,\infty)\), and \(\lambda\) is a nonzero finite Lebesgue measure on the unit sphere \(S^{n-1} \). Such a measure can be produced by a variational formula of \(\tilde{V}_{G, \lambda}(K)\) (the general dual volume of \(K\) with respect to \(\lambda\)) under the perturbations of \(K\) by the Musielak-Orlicz addition defined via the function \(\Psi \). The Musielak-Orlicz-Gauss image problem contains many intensively studied Minkowski-type problems and the recent Gauss image problem as its special cases. Under the condition that \(G\) is decreasing on its second variable, the existence of solutions to this problem is established.Vectorial analogues of Cauchy's surface area formulahttps://zbmath.org/1540.520062024-09-13T18:40:28.020319Z"Hug, Daniel"https://zbmath.org/authors/?q=ai:hug.daniel"Schneider, Rolf"https://zbmath.org/authors/?q=ai:schneider.rolf-gCauchy's surface area formula says that for a \(d\)-dimensional convex body \(K\), the mean value of the measure of an orthogonal projection of \(K\) onto a hyperplane, taken over all hyperplanes through the origin, is proportional to the surface area of \(K\). This follows (roughly) upon integrating the contribution \(|\cos \theta\; dA|\) of a patch to get (twice) the measure of a specific projection, integrating this to get the mean, and changing the order of integration.
The authors do something similar to give an expression for the integral over \(S^{d-1}\) of the moment vectors of the projections in terms of the vector quantities \(q_1(K)\) and \(\Upsilon_1(K)\). (The vector \(q_1(K)\) is a coefficient in the Steiner-like vector polynomial
\[
z_{d+1}(K+\lambda B^d) = \sum_{j=0}^d \binom{d}{j}\lambda^j q_j(K)
\]
for the moment vector \(z_{d+1}\).)
Reviewer: Robert Dawson (Halifax)Generation of random pseudo-line arrangementshttps://zbmath.org/1540.520072024-09-13T18:40:28.020319Z"Roch, Sandro M."https://zbmath.org/authors/?q=ai:roch.sandro-mIn this nice overview, the author presents some basics about pseudoline arrangements in the real (projective) plane, i.e. arrangements of smooth simple plane curves such that each pair of curves crosses transversally at exactly one point. After a nice introduction to the subject, including a question about the stretchability of pseudoline arrangements (which is NP-hard) or about different techniques we can use to understand such arrangements (such as wiring diagrams), the author addresses a question about random pseudoline arrangements and how to use Markov chains to construct such random arrangements with some specific properties. The main result that the author presents is devoted to the rapidly mixing Markov chains that flip only random triangles in general random pseudoline arrangements, and for details we refer to the last theorem in the note.
Reviewer: Piotr Pokora (Kraków)Line transversals to disjoint diskshttps://zbmath.org/1540.520082024-09-13T18:40:28.020319Z"González-Arreola, E."https://zbmath.org/authors/?q=ai:gonzalez-arreola.edgar"Jerónimo-Castro, J."https://zbmath.org/authors/?q=ai:jeronimo-castro.jesusAuthors' abstract: The main purpose of this paper is to prove the following result: Let \(\mathcal {F}\) be a \(\sqrt 2\)-disjoint finite family of disks with the property that every four of them have a common line transversal. Then there exists a line transversal to all members of \(\mathcal {F}\). Moreover, we prove that the number \(\sqrt 2 \) is sharp.
Reviewer: Jan Kurek (Lublin)The \(L_p\) chord Minkowski problem for negative \(p\)https://zbmath.org/1540.520092024-09-13T18:40:28.020319Z"Li, Yuanyuan"https://zbmath.org/authors/?q=ai:li.yuanyuan.2The chord measures and the Minkowski problems associated with chord measures are introduced by Lutwak, Xi, Yang and Zhang, and the \(L_p\) chord Minkowski problem also posed by them. \textit{D. Xi} et al. [Adv. Nonlinear Stud. 23, Article ID 20220041, 22 p. (2023; Zbl 07655242)] solved the \(L_p\) chord Minkowski problem for \(p>1\) and for \(0<p<1\). Xi, Guo,and Zhao solved the \(L_p\) chord Minkowski problem when \(0\le p<1\). Hu, Huang, Lu, and Wang gave a new existence result of solutions to this problem with smooth even measure for \(p>-n\) with \(p\neq0\). This paper solves the \(L_p\) chord Minkowski problem in the case of discrete measures whose supports are in general position for \(p<0\) and \(q>0\).
Reviewer: Wenxue Xu (Chongqing)Orlicz geominimal surface areashttps://zbmath.org/1540.520102024-09-13T18:40:28.020319Z"Zhao, Chang-Jian"https://zbmath.org/authors/?q=ai:zhao.changjianSummary: The main purpose of the present article is to introduce a new concept and call it \textit{Orlicz mixed geominimal surface area} \(G_\varphi (K_1, \dots, K_n)\) of \(n\) convex bodies \(K_1, \dots, K_n \), which obeys classical basic properties. The new affine geometric quantity in special case yields Petty's geominimal surface area \(G(K)\), Lutwak's \(p\)-geominimal surface area \(G_p(K)\) and the newly established Orlicz geominimal surface area \(G_\varphi(K)\). The Orlicz mixed geominimal surface area inequality is established, which in special case yields Petty's geominimal surface area inequality, Lutwak's \(p\)-geominimal surface area inequality and Orlicz geominimal surface area inequality, respectively. Moreover, the related concepts and inequalities of \(L_p\)-mixed geominimal surface area are also derived here.From the Brunn-Minkowski inequality to a class of generalized Poincaré-type inequalities for torsional rigidityhttps://zbmath.org/1540.520112024-09-13T18:40:28.020319Z"Fang, Niufa"https://zbmath.org/authors/?q=ai:fang.niufa"Hu, Jinrong"https://zbmath.org/authors/?q=ai:hu.jinrong"Zhao, Leina"https://zbmath.org/authors/?q=ai:zhao.leinaTorsional rigidity of a domain in \(\mathbb{R}^n\) can defined via the solution of a boundary value problem, or variationally. The latter has been exploited by Colesanti and Fimiani to obtain a Brunn-Minkowski inequality for the torsional rigidity of convex combinations of convex domains and to define a torsional measure of a convex domain. In this paper, the authors show that the Brunn-Minkowski inequality implies a Poincaré-type inequality with respect to the torsional measure on smooth convex domain in \(\mathbb{R}^n\) with strictly positive curvature. The set-up looks promising to extend to other problems in potential theory and analysis.
Reviewer: Alina Stancu (Montréal)Surface areas of equifacetal polytopes inscribed in the unit sphere \(\mathbb{S}^2\)https://zbmath.org/1540.520122024-09-13T18:40:28.020319Z"Freeman, Nicolas"https://zbmath.org/authors/?q=ai:freeman.nicolas"Hoehner, Steven"https://zbmath.org/authors/?q=ai:hoehner.steven-d"Ledford, Jeff"https://zbmath.org/authors/?q=ai:ledford.jeff"Pack, David"https://zbmath.org/authors/?q=ai:pack.david"Walters, Brandon"https://zbmath.org/authors/?q=ai:walters.brandonGiven an integer \(K\) greater than 4, let \({\mathcal I}_K\) stand for the set of all convex polytopes with at most \(K\) distinct vertices on the unit sphere \({\mathbb S}^2\). Also, denote the corresponding subset of polytopes with congruent isosceles or congruent equilateral triangular facets by \({\mathcal M}_K\). The main results yield the surface area maximizers in \({\mathcal M}_7\) and \({\mathcal M}_8\).
Reviewer: S. S. Kutateladze (Novosibirsk)Analytic aspects of the dilation inequality for symmetric convex sets in Euclidean spaceshttps://zbmath.org/1540.520132024-09-13T18:40:28.020319Z"Tsuji, Hiroshi"https://zbmath.org/authors/?q=ai:tsuji.hiroshi.1Summary: We discuss an analytic form of the dilation inequality with respect to a probability measure for symmetric convex sets in Euclidean spaces, which is a counterpart of analytic aspects of Cheeger's isoperimetric inequality. We show that the dilation inequality for symmetric convex sets is equivalent to a certain bound of the relative entropy for even quasi-convex functions, which is close to the logarithmic Sobolev inequality or Cramér-Rao inequality. As corollaries, we investigate the reverse Shannon inequality, logarithmic Sobolev inequality, Kahane-Khintchine inequality, deviation inequality and isoperimetry. We also give new probability measures satisfying the dilation inequality for symmetric convex sets via bounded perturbations and tensorization.Many neighborly sphereshttps://zbmath.org/1540.520142024-09-13T18:40:28.020319Z"Novik, Isabella"https://zbmath.org/authors/?q=ai:novik.isabella"Zheng, Hailun"https://zbmath.org/authors/?q=ai:zheng.hailunA simplicial complex on \(n\) vertices is \emph{\(s\)-neighborly} if it has the same \((s-1)\)-skeleton as the \((n-1)\)-simplex on the same vertex set. Inspired by a conjecture by \textit{G. Kalai} [Discrete Comput. Geom. 3, No. 1--2, 1--14 (1988; Zbl 0631.52009)], the main result of this paper is an explicit construction, using \textit{Kalai's squeezed balls} (also from Kalai [loc. cit.]), to show that if \(\mathrm{sn}(d,n)\) denotes the number of \(\lfloor d/2\rfloor\)-neighborly simplicial \((d-1)\)-spheres with \(n\) labeled vertices, then \(\mathrm{sn}(d,n) \geq 2^{\Omega(n^{\lfloor (d-1)/2\rfloor})}\) for all \(n\geq 5\).
Reviewer: Geir Agnarsson (Fairfax)Chiral polytopes whose smallest regular cover is a polytopehttps://zbmath.org/1540.520152024-09-13T18:40:28.020319Z"Cunningham, Gabe"https://zbmath.org/authors/?q=ai:cunningham.gabeAn abstract polytope is defined as a graded complex satisfying the condition of diamond condition and connectivity.
A flag is a maximal flag in the graded complex. For a polytope of dimension \(n\) the flag \(f=(f_0, \dots, f_n)\) is of length \(n+1\). For \(0\leq i \leq n\), \(\sigma_i(f)\) is the unique flag adjacent to \(f\) on position \(i\).
If \({\mathcal P}\) and \({\mathcal Q}\) are two abstract polytopes, then we say that \({\mathcal P}\) covers \({\mathcal Q}\) if there is a function \(\pi: {\mathcal P}\to {\mathcal Q}\) which preserves the partial order, the rank of each face and such that \(\pi\circ \sigma_i = \sigma_i \circ \pi\).
An automorphism is a bijection of \({\mathcal P}\). The group acts on the flags. If there is just one orbit of flags then the automorphism group is a quotient of a string Coxeter group. The group is chiral if there are two orbits of flags such that for all \(i\) and flag \(f\) then \(f\) and \(\sigma_i(f)\) are in different orbits.
The authors then study the condition under which the smallest regular cover of a chiral abstract polytope is a polytope.
Reviewer: Mathieu Dutour Sikirić (Zagreb)Perfect and almost perfect homogeneous polytopeshttps://zbmath.org/1540.520162024-09-13T18:40:28.020319Z"Berestovskiĭ, V. N."https://zbmath.org/authors/?q=ai:berestovskii.valerii-nikolaevich"Nikonorov, Yu. G."https://zbmath.org/authors/?q=ai:nikonorov.yurii-gennadevichA Euclidean polytope of arbitrary dimension is called vertex-transitive when its symmetry group acts transitively on its vertex set. Vertex-transitive polytopes form a superset of the regular polytopes (whose symmetry group acts transitively on the set of flags) and of the semi-regular polytopes (that, on top of being vertex-transitive, have regular facets).
In this article, two subsets of the vertex-transitive polytopes are considered, the perfect and almost perfect polytopes: a vertex-transitive polytope is perfect when the only degree two hypersurface \(S\) through all of its vertices is its circumsphere. Almost-perfect polytopes are defined using a slightly stronger condition on \(S\). The article studies how perfect and almost perfect polytopes relate to regular and semi-regular polytopes. In particular, the perfect and almost perfect regular polytopes are classified and, except for an explicit list of exceptions, so are the perfect and almost perfect semi-regular polytopes.
Reviewer: Lionel Pournin (Paris)A simple introduction to higher order liftings for binary problemshttps://zbmath.org/1540.520172024-09-13T18:40:28.020319Z"Jarre, Florian"https://zbmath.org/authors/?q=ai:jarre.florianThe \textit{max-cut-polytope} \(\boldsymbol{\mathrm{MC}}\subseteq {\mathbb{R}}^{n^2}\) is defined as the convex hull of all \(\tilde{x}\tilde{x}^T\) where \(\tilde{x}\in \{\pm 1\}^n\) is viewed as a column vector. This paper presents a short, simple and self-contained proof of the fact that the \(n\)-th lifting of the max-cut-polytope is exact. This proof re-derives the known observation that \(\boldsymbol{\mathrm{MC}}\) is the projection of a higher-dimensional regular simplex which coincides with the \(n\)-th semidefinite lifting. The paper concludes with an extension to reduce the dimension of higher order liftings and to include all the linear equality and inequality constraints. All the results presented in this paper are well-known and some relations to earlier work are detailed in the final section.
Reviewer: Geir Agnarsson (Fairfax)Two enriched poset polytopeshttps://zbmath.org/1540.520182024-09-13T18:40:28.020319Z"Okada, Soichi"https://zbmath.org/authors/?q=ai:okada.soichi"Tsuchiya, Akiyoshi"https://zbmath.org/authors/?q=ai:tsuchiya.akiyoshiIn a celebrated paper, \textit{R. P. Stanley} [Discrete Comput. Geom. 1, 9--23 (1986; Zbl 0595.52008)] introduced the \textit{order polytope} \(O(P)\) and the \textit{chain polytope} \(C(P)\) of a finite partially order set (poset) \(P\), which are \(0/1\)-polytopes whose defining inequalities come from the order relations and chains of \(P\), respectively. He also provided unimodular triangulations for both polytopes and constructed a piece-wise linear map which yields a bijection between \(m \, O(P) \cap\mathbf{Z}^P\) and \(m \, C(P) \cap \mathbf{Z}^P\), for any positive integer \(m\); consequently, \(O(P)\) and \(C(P)\) have the same Ehrhart polynomial.
Motivated by work on enriched \(P\)-partitions, \textit{H. Ohsugi} and \textit{A. Tsuchiya} [Isr. J. Math. 237, No. 1, 485--500 (2020; Zbl 1453.52013); Eur. J. Math. 7, No. 1, 48--68 (2021; Zbl 1462.52020)] recently introduced enriched versions of order and chain polytopes, which are \(0, \pm 1\)-polytopes; enriched chain polytopes were independently introduced in [\textit{F. Kohl} et al., Discrete Comput. Geom. 64, No. 2, 427--452 (2020; Zbl 1454.52012)] as unconditional chain polytopes. Ohsugi and Tsuchiya proved that the enriched order and chain polytopes also have the same Ehrhart polynomial, and the paper under review extends Stanley's transfer map to prove this bijectively. The authors also construct unimodular triangulations of the enriched order and chain polytopes.
Reviewer: Matthias Beck (San Francisco)Rational Ehrhart theoryhttps://zbmath.org/1540.520192024-09-13T18:40:28.020319Z"Beck, Matthias"https://zbmath.org/authors/?q=ai:beck.matthias"Elia, Sophia"https://zbmath.org/authors/?q=ai:elia.sophia"Rehberg, Sophie"https://zbmath.org/authors/?q=ai:rehberg.sophieSummary: The Ehrhart quasipolynomial of a rational polytope \textsf{P} encodes fundamental arithmetic data of \textsf{P}, namely, the number of integer lattice points in positive integral dilates of \textsf{P}. Ehrhart quasipolynomials were introduced in the 1960s, they satisfy several fundamental structural results and have applications in many areas of mathematics and beyond. The enumerative theory of lattice points in rational (equivalently, real) dilates of rational polytopes is much younger, starting with work by \textit{E. Linke} [J. Comb. Theory, Ser. A 118, No. 7, 1966--1978 (2011; Zbl 1234.52010)], \textit{V. Baldoni} et al. [Mathematika 59, No. 1, 1--22 (2013; Zbl 1260.05006)], and \textit{A. Stapledon} [J. Comb. Theory, Ser. A 151, 51--60 (2017; Zbl 1370.52028)]. We introduce a generating-function \textit{ansatz} for rational Ehrhart quasipolynomials, which unifies several known results in classical and rational Ehrhart theory. In particular, we define \(\gamma\)-rational Gorenstein polytopes, which extend the classical notion to the rational setting and encompass the generalized reflexive polytopes studied by \textit{M. H. J. Fiset} and \textit{A. M. Kasprzyk} [Electron. J. Comb. 15, No. 1, Research Paper N18, 4 p. (2008; Zbl 1163.05304)] and \textit{A. M. Kasprzyk} and \textit{B. Nill} [Electron. J. Comb. 19, No. 3, Research Paper P9, 18 p. (2012; Zbl 1258.52008)].Families of polytopes with rational linear precision in higher dimensionshttps://zbmath.org/1540.520202024-09-13T18:40:28.020319Z"Davies, Isobel"https://zbmath.org/authors/?q=ai:davies.isobel"Duarte, Eliana"https://zbmath.org/authors/?q=ai:duarte.eliana"Portakal, Irem"https://zbmath.org/authors/?q=ai:portakal.irem"Sorea, Miruna-Ştefana"https://zbmath.org/authors/?q=ai:sorea.miruna-stefanaThe paper is motivated by geometric modelling and in particular, by the construction of blending functions. As a generalization of Bézier curves and surfaces, \textit{R. Krasauskas} [Adv. Comput. Math. 17, No. 1--2, 89--113 (2002; Zbl 0997.65027)] defined \textit{toric patches}, which have a lattice polytope \(P \subseteq \mathbb{R}^d\) as domain and whose blending functions are constructed from the set \(\mathcal{A} = P \cap \mathbb{Z}^d\) of lattice points, the facet defining inequalities of~\(P\), and a weight vector \(w \in \mathbb{R}_{>0}^\mathcal{A}\). A desirable property for geometric modelling is what is called \textit{linear precision}, meaning that the blending functions can replicate any affine function on the domain. It was noted in [\textit{L. D. Garcia-Puente} and \textit{F. Sottile}, Adv. Comput. Math. 33, No. 2, 191--214 (2010; Zbl 1193.65018)] that this property strongly depends on the defining set \(\mathcal{A}\), and that pairs \((P,w)\) with linear precision can be conveniently characterized.
A variant of this notion called \textit{rational linear precision} (RLP) was defined in [Garcia-Puente and Sottile, loc. cit.] and basically asks for a reparametrisation of the underlying toric variety \(X_{\mathcal{A},w}\) by rational functions (quotients of polynomial functions). The classification of pairs \((P,w)\) with RLP is only established in dimension \(d=2\) so far, and for \(d \geq 3\) it remains an open problem of interest in the community. The authors contribute to this problem by describing a large novel family of lattice polytopes~\(P\) and associated weights~\(w\) with RLP (Theorem 4.1). The family is constructed from what the authors call \textit{multinomial staged trees}, building on the equivalence proved in [Garcia-Puente and Sottile, loc. cit.] that a pair \((P,w)\) has RLP if and only if the associated projective toric variety \(X_{\mathcal{A},w}\) has rational maximum likelihood estimator. This allows to use methods from Algebraic Statistics to study the RLP property.
Reviewer: Matthias Schymura (Cottbus)Bounds on area involving lattice sizehttps://zbmath.org/1540.520212024-09-13T18:40:28.020319Z"Soprunova, Jenya"https://zbmath.org/authors/?q=ai:soprunova.evgeniaThe \textit{lattice size} of a convex body \(P \subset\mathbf{R}^2\) with respect to a set \(X \subset\mathbf{R}^2\) is the smallest \(t \in\mathbf{R}_{ \ge 0 }\) such that a unimodular image of \(P\) is contained in \(t X\); here a \textit{unimmodular map} is a transformation in SL\(_2(\mathbf{Z})\) followed by a lattice translation. The lattice size coincides with the \textit{lattice width} of \(P\) when choosing \(X = [0,1] \times\mathbf{R}\).
The paper under review proves that the area of a lattice polygon is bounded below by \(\frac{1}{2}\) times its lattice size with respect to either the unimodular triangle conv\(\{ (0,0, (1,0), (0,1) \}\) and the unit square \([0,1]^2\), and that these bounds are sharp. The author also classifies inclusion-minimal lattice polygons with fixed lattice size with respect to the unit square.
Reviewer: Matthias Beck (San Francisco)\(\mathrm{SL}(n)\) contravariant function-valued valuations on polytopeshttps://zbmath.org/1540.520222024-09-13T18:40:28.020319Z"Tang, Zhongwen"https://zbmath.org/authors/?q=ai:tang.zhongwen"Li, Jin"https://zbmath.org/authors/?q=ai:li.jin"Leng, Gangsong"https://zbmath.org/authors/?q=ai:leng.gangsongSummary: We present a complete classification of \(\mathrm{SL}(n)\) contravariant, \(C (\mathbb{R}^n \setminus \{o\})\)-valued valuations on polytopes, without any additional assumptions. It extends the previous results of the second author \textit{J. Li} [Int. Math. Res. Not. 2020, No. 22, 8197--8233 (2020; Zbl 1459.52011)] which have a good connection with the \(L_p\) and Orlicz Brunn-Minkowski theory. Additionally, our results deduce a complete classification of \(\mathrm{SL}(n)\) contravariant symmetric-tensor-valued valuations on polytopes.Finding weakly simple closed quasigeodesics on polyhedral sphereshttps://zbmath.org/1540.520232024-09-13T18:40:28.020319Z"Chartier, Jean"https://zbmath.org/authors/?q=ai:chartier.jean"de Mesmay, Arnaud"https://zbmath.org/authors/?q=ai:de-mesmay.arnaud\textit{A. V. Pogorelov} [Mat. Sb., Nov. Ser. 25(67), 275--306 (1949; Zbl 0041.08902)] proved the existence of a simple closed quasigeodesic on any convex polyhedron. The authors consider a similar problem for so-called polyhedral spheres. Such spheres are glued from a finite set of Euclidean polygons and are not a priori embedded in \(\mathbb{R}^3\).
Theorem 1.1 states that for a polyhedral sphere \(S\) there exists a weakly simple closed quasigeodesic of length at most \(M\). Here \(M\) denotes the edge-sum of \(S\), i.e., the sum of the lengths of the edges of an iterated barycentric subdivision of a triangulation of \(S\), and a weakly simple curve is a limit of simple curves on \(S\). The authors prove in Theorem 1.2 that it is possible to construct a weakly simple closed quasigeodesic, in exponential time with respect to \(n\) and \([M/h]\), where \(n\) is the number of vertices of \(S\) and \(h\) is the smallest altitude over all triangles of some triangulation of \(S\). This theorem answers a question suggested by \textit{E. D. Demaine} et al. [LIPIcs -- Leibniz Int. Proc. Inform. 164, Article 33, 13 p. (2020; Zbl 1533.68358)].
Reviewer: Samyon R. Nasyrov (Kazan)Bounds on soft rectangle packing ratioshttps://zbmath.org/1540.520242024-09-13T18:40:28.020319Z"Brecklinghaus, Judith"https://zbmath.org/authors/?q=ai:brecklinghaus.judith"Brenner, Ulrich"https://zbmath.org/authors/?q=ai:brenner.ulrich"Kiss, Oliver"https://zbmath.org/authors/?q=ai:kiss.oliver.1Summary: We examine rectangle packing problems where only the areas \(a_1, \dots, a_n\) of the rectangles to be packed are given while their aspect ratios may be chosen from a given interval \([\frac{1}{\gamma}, \gamma]\). In particular, we ask for the smallest possible size of a rectangle \(R\) such that, under these constraints, any collection \(a_1, \dots, a_n\) of rectangle areas of total size 1 can be packed into \(R\). As for standard square packing problems, which are contained as a special case for \(\gamma = 1\), this question leads us to three different answers, depending on whether the aspect ratio of \(R\) is given or whether we may choose it either with or without knowing the areas \(a_1, \dots, a_n\). Generalizing known results for square packing problems, we provide upper and lower bounds for the size of \(R\) with respect to all three variants of the problem, which are tight at least for larger values of \(\gamma\). Moreover, we show how to improve these bounds on the size of \(R\) if we restrict ourselves to instances where the largest element in \(a_1, \dots, a_n\) is bounded.Local groups in Delone setshttps://zbmath.org/1540.520252024-09-13T18:40:28.020319Z"Dolbilin, Nikolay"https://zbmath.org/authors/?q=ai:dolbilin.nikolai-pSummary: We prove that in an arbitrary Delone set \(X\) in the three-dimensional space, the subset \(X_6\) of all points from \(X\) at which the local group has no rotation axis of order larger than 6 is also a Delone set. Here, under the local group at point \(x \in X\) we mean the symmetry group \(S_x(2R)\) of the cluster \(C_x(2R)\) of \(x\) with radius \(2R\), where \(R\) is the radius of the largest ball free of points of \(X\) (according to Delone's empty spheretheory). The main result seems to be the first rigorously proved statement for absolutely generic Delone sets which implies substantial statements for Delone sets with strong crystallographic restrictions. For instance, an important observation of Shtogrin on the boundedness of local groups in Delone sets with equivalent \(2R\)-clusters immediately follows from the main result. Further, we propose a crystalline kernel conjecture and its two weaker versions. According to the crystalline kernel conjecture, in an arbitrary Delone set, points with locally crystallographic axes only (i.e., of order 1, 2, 3, 4, or 6) inevitably constitute the essential part of the set. These conjectures significantly generalize the famous crystallography statement on the impossibility of a (global) fivefold symmetry in a three-dimensional lattice.
For the entire collection see [Zbl 1470.65003].Quotient graphs of symmetrically rigid frameworkshttps://zbmath.org/1540.520262024-09-13T18:40:28.020319Z"Dewar, Sean"https://zbmath.org/authors/?q=ai:dewar.sean"Grasegger, Georg"https://zbmath.org/authors/?q=ai:grasegger.georg"Kastis, Eleftherios"https://zbmath.org/authors/?q=ai:kastis.eleftherios"Nixon, Anthony"https://zbmath.org/authors/?q=ai:nixon.anthonySummary: A natural problem in combinatorial rigidity theory concerns the determination of the rigidity or flexibility of bar-joint frameworks in \(\mathbb{R}^d\) that admit some non-trivial symmetry. When \(d=2\) there is a large literature on this topic. In particular, it is typical to quotient the symmetric graph by the group and analyse the rigidity of symmetric, but otherwise generic frameworks, using the combinatorial structure of the appropriate group-labelled quotient graph. However, mirroring the situation for generic rigidity, little is known combinatorially when \(d\geq 3\). Nevertheless in the periodic case, a key result of \textit{C. S. Borcea} and \textit{I. Streinu} in [Bull. Lond. Math. Soc. 43, No. 6, 1093--1103 (2011; Zbl 1235.52031)] characterises when a quotient graph can be lifted to a rigid periodic framework in \(\mathbb{R}^d\). We develop an analogous theory for symmetric frameworks in \(\mathbb{R}^d\). The results obtained apply to all finite and infinite 2-dimensional point groups, and then in arbitrary dimension they concern a wide range of infinite point groups, sufficiently large finite groups and groups containing translations and rotations. For the case of finite groups we also derive results concerning the probability of assigning group labels to a quotient graph so that the resulting lift is symmetrically rigid in \(\mathbb{R}^d\).On Yuzvinsky's lattice sheaf cohomology for hyperplane arrangementshttps://zbmath.org/1540.520272024-09-13T18:40:28.020319Z"Mücksch, Paul"https://zbmath.org/authors/?q=ai:mucksch.paulSummary: We establish the relationship between the cohomology of a certain sheaf on the intersection lattice of a hyperplane arrangement introduced by Yuzvinsky and the cohomology of the coherent sheaf on punctured affine space, respectively projective space associated to the module of logarithmic vector fields along the arrangement. Our main result gives a Künneth formula connecting the cohomology theories, answering a question by Yoshinaga. This, in turn, provides a characterization of the projective dimension of the module of logarithmic vector fields and yields a new proof of Yuzvinsky's freeness criterion. Furthermore, our approach affords a new formulation of Terao's freeness conjecture and a more general problem.Freeness for restriction arrangements of the extended Shi and Catalan arrangementshttps://zbmath.org/1540.520282024-09-13T18:40:28.020319Z"Nakashima, Norihiro"https://zbmath.org/authors/?q=ai:nakashima.norihiro"Tsujie, Shuhei"https://zbmath.org/authors/?q=ai:tsujie.shuheiSummary: The extended Shi and Catalan arrangements are well investigated arrangements. In this paper, we prove that the cone of the extended Catalan arrangement of type A is always hereditarily free, while we determine the dimension in which the cone of the extended Shi arrangement of type A is hereditarily free. For this purpose, using digraphs, we define a class of arrangements which is closed under restriction, and which contains the extended Shi and Catalan arrangements. We also characterize the freeness for the cone of this arrangement by graphical conditions.On \(k\)-neighborly reorientations of oriented matroidshttps://zbmath.org/1540.520292024-09-13T18:40:28.020319Z"Hernández-Ortiz, Rangel"https://zbmath.org/authors/?q=ai:hernandez-ortiz.rangel"Knauer, Kolja"https://zbmath.org/authors/?q=ai:knauer.kolja-b"Montejano, Luis Pedro"https://zbmath.org/authors/?q=ai:montejano.luis-pedroSummary: We study the existence and the number of \(k\)-neighborly reorientations of an oriented matroid. This leads to \(k\)-variants of McMullen's problem and Roudneff's conjecture, the case \(k = 1\) being the original statements. Adding to results of Larman and García-Colín, we provide new bounds on \(k\)-McMullen's problem and prove the conjecture for several ranks and \(k\) by computer. Further, we show that \(k\)-Roudneff's conjecture for fixed rank and \(k\) reduces to a finite case analysis. As a consequence we prove the conjecture for odd rank \(r\) and \(k = \frac{r-1}{2}\) as well as for rank 6 and \(k = 2\) with the aid of the computer.A generalized combinatorial Ricci flow on surfaces of finite topological typehttps://zbmath.org/1540.531222024-09-13T18:40:28.020319Z"Ba, Te"https://zbmath.org/authors/?q=ai:ba.te"Li, Shengyu"https://zbmath.org/authors/?q=ai:li.shengyu"Xu, Yaping"https://zbmath.org/authors/?q=ai:xu.yapingSummary: We introduce a generalized combinatorial Ricci flow on surfaces of finite topological type. Using a Lyapunov function, we prove that the flow exists for all time and converges to a circle pattern metric on surfaces with prescribed curvatures. This suggests an algorithm to find circle patterns on surfaces with obtuse exterior intersection angles. As a comparison, this flow has the advantage of accelerating the convergence rate.Combinatorial curvature flows for generalized circle packings on surfaces with boundaryhttps://zbmath.org/1540.531282024-09-13T18:40:28.020319Z"Xu, Xu"https://zbmath.org/authors/?q=ai:xu.xu"Zheng, Chao"https://zbmath.org/authors/?q=ai:zheng.chaoThe authors study the deformations of generalized circle packings on an ideally triangulated surface with boundary.
Let \(\Sigma\) be a closed compact surface. A circle packing on \(\Sigma\) is a collection of disjoint open disks with respect to a hyperbolic metric (or more generally, a projective structure) on \(\Sigma\). The combinatorics of the tangency between disks in a circle packing can be encoded in a graph \(\mathcal{G}\) on \(\Sigma\). The classical rigidity theorem of Koebe-Andreev-Thurston states that if the graph \(\mathcal{G}\) is a triangulation of \(\Sigma\), then there exists a unique hyperbolic structure on \(\Sigma\) supporting a circle packing whose combinatorics is encoded by \(\mathcal{G}\), see [\textit{W. P. Thurston}, The geometry and topology of three-manifolds. Providence, RI: American Mathematical Society (AMS) (2022; Zbl 1507.57005)]. More generally, one can consider circle patterns, where disks are allowed to intersect, and the intersecting angles are recorded as weights on the graph \(\mathcal{G}\). In this case a similar rigidity result holds.
Even more generally, \textit{R. Guo} and \textit{F. Luo} [Geom. Topol. 13, No. 3, 1265--1312 (2009; Zbl 1160.52012)] considered generalized circle packing metrics associated with a triangulation \(\mathcal{G}\) and weights on the edges of \(\mathcal{G}\). Instead of intersecting angles between circles, the weights may encode lengths or angles depending on the type.
In this paper, the authors consider specific generalized circle packing metrics of type \((-1,-1,-1)\). In this case, the combinatorial data is realized as the decomposition of a hyperbolic surface \(\Sigma\) with geodesic boundary into right-angled hyperbolic hexagons, where each side of the hexagons either lies on the boundary of \(\Sigma\) or is a segment connecting two boundary components.
To deform such structures, the authors introduce several combinatorial analogues of curvature flows from differential geometry, namely the combinatorial Ricci flow, the combinatorial Calabi flow, and the fractional combinatorial Calabi flow. The authors prove long-time existence and global exponential convergence of solutions for the first two flows with general initial values. For the last flow, the authors show existence and convergence for certain initial values, and they conjecture that the result holds in general.
Reviewer: Yongquan Zhang (Stony Brook)Topology of the Grünbaum-Hadwiger-Ramos problem for mass assignmentshttps://zbmath.org/1540.550092024-09-13T18:40:28.020319Z"Blagojević, Pavle V. M."https://zbmath.org/authors/?q=ai:blagojevic.pavle-v-m"Loperena, Jaime Calles"https://zbmath.org/authors/?q=ai:loperena.jaime-calles"Crabb, Michael C."https://zbmath.org/authors/?q=ai:crabb.michael-c"Blagojević, Aleksandra S. Dimitrijević"https://zbmath.org/authors/?q=ai:dimitrijevic-blagojevic.aleksandra-sAuthors' abstract: In this paper, motivated by recent work of \textit{P. Schnider} [LIPIcs -- Leibniz Int. Proc. Inform. 129, Article 56, 15 p. (2019; Zbl 07559256)] and \textit{I. Axelrod-Freed} and \textit{P. Soberón} [``Bisections of mass assignments using flags of affine spaces'', Preprint, \url{arXiv:2109.13106}], we study an extension of the classical Grünbaum-Hadwiger-Ramos mass partition problem to mass assignments. Using the Fadell-Husseini index theory we prove that for a given family of \(j\) mass assignments \(\mu_1,\ldots,\mu_j\) on the Grassmann manifold \(G_{\ell}\big(\mathbb{R}^d\big)\) and a given integer \(k\geq 1\) there exist a linear subspace \(L\in G_{\ell}\big(\mathbb{R}^d\big)\) and \(k\) affine hyperplanes in \(L\) that equipart the masses \(\mu_1^L, \ldots, \mu_j^L\) assigned to the subspace \(L\), provided that \(d\geq j + (2^{k-1}-1)2^{\lfloor\log_2 j\rfloor}\).
Reviewer: Mircea Balaj (Oradea)Peeling random planar maps. École d'Été de Probabilités de Saint-Flour XLIX -- 2019https://zbmath.org/1540.600012024-09-13T18:40:28.020319Z"Curien, Nicolas"https://zbmath.org/authors/?q=ai:curien.nicolasPublisher's description: These Lecture Notes provide an introduction to the study of those discrete surfaces which are obtained by randomly gluing polygons along their sides in a plane. The focus is on the geometry of such random planar maps (diameter, volume growth, scaling and local limits...) as well as the behavior of statistical mechanics models on them (percolation, simple random walks, self-avoiding random walks...).
A ``Markovian'' approach is adopted to explore these random discrete surfaces, which is then related to the analogous one-dimensional random walk processes. This technique, known as ``peeling exploration'' in the literature, can be seen as a generalization of the well-known coding processes for random trees (e.g. breadth first or depth first search). It is revealed that different types of Markovian explorations can yield different types of information about a surface.
Based on an École d'Été de Probabilités de Saint-Flour course delivered by the author in 2019, the book is aimed at PhD students and researchers interested in graph theory, combinatorial probability and geometry. Featuring open problems and a wealth of interesting figures, it is the first book to be published on the theory of random planar maps.Norms of structured random matriceshttps://zbmath.org/1540.600102024-09-13T18:40:28.020319Z"Adamczak, Radosław"https://zbmath.org/authors/?q=ai:adamczak.radoslaw"Prochno, Joscha"https://zbmath.org/authors/?q=ai:prochno.joscha"Strzelecka, Marta"https://zbmath.org/authors/?q=ai:strzelecka.marta"Strzelecki, Michał"https://zbmath.org/authors/?q=ai:strzelecki.michalFor \( m,n\in\mathbb{N}\) let \(X = (X_{i, j})_{1\leq i \leq m,1\leq j\leq n}\) be a random matrix, \(A = (a_{i, j})_{ 1\leq i \leq m,1\leq j \leq n}\) be a deterministic matrix with real numbers as entries, and \(X_{A}= X\circ A\) the corresponding Hadamard product that is a structured random matrix. Bounds of the expected operator norm of \(X_{A}, \) considered as a random operator between \(\ell^{n}_{p}\) and \(\ell^{n}_{q}, 1\leq p, q \leq \infty,\) are studied. They are of optimal order and can be expressed in terms of the entries of the matrix \(A\). The understanding of such expressions and related quantities is relevant in the study of the worst-case error of optimal algorithms which are based on random information in function approximation problems, see [\textit{D. Krieg} and \textit{M. Ullrich}, Found. Comput. Math. 21, No. 4, 1141--1151 (2021; Zbl 1481.41014)], or the quality of random information for the recovery of vectors from an \(\ell\)-ellipsoid where the radius of the optimal information is given by Gelfand numbers of a diagonal operator, see [\textit{A. Hinrichs} et al., J. Approx. Theory 293, Article ID 105919, 19 p. (2023; Zbl 1531.52005)].
For optimal bounds up to logarithmic terms when \(X\) has i. i. d. Gaussian entries, independent mean-zero bounded entries are deduced. Indeed, if \(D_{1}= || A\circ A: \ell^{n}_{p/2} \rightarrow\ell^{m}_{q/2} ||^{1/2}\) and \(D_{2}= || (A\circ A)^{T} :\ell^{m}_{q^{*}/2} \rightarrow\ell^{n}_{p^{*}/2} ||^{1/2},\) where \(p^{*}\) denotes the Hölder conjugate of \(p,\) then
\[
D_{1} + D_{2} \lesssim \mathbb{E} ||X_{A}: l^{n}_{p} \rightarrow l^{m}_{q}||\lesssim (\ln( n) )^{1/p^{*}} (\ln( m))^{1/q}[ \sqrt{ \ln (mn)} D_{1} + \sqrt{ \ln (n)} D_{2}].
\]
When one deals with particular ranges of \(p,q\) the precise order of the expected norm up to constants is determined.
The core of the paper are two conjectures. Conjecture 1 deals with estimates of \(\mathbb{E}||X_{A}:\ell^{n}_{p}\rightarrow\ell^{n}_{q}||.\) Conjecture 2 states that the boundedness of the linear operator \(X_{A}\) given by an infinite dimensional matrix \(A\) is equivalent to the fact that \(A\circ A\) defines a bounded linear operator between \(\ell_{p/2} (\mathbb{N})\) and \(\ell_{q/2} (\mathbb{N}),\) \((A\circ A)^{T}\) defines a bounded linear operator between \(\ell_{q^{*}/2} (\mathbb{N})\) and \(\ell_{p^{*}/2} (\mathbb{N}),\) as well as some extra conditions depending on the norms of the infinite row and column vectors of \(A\) and the values of \( p\) and \(q\) hold.
In addition to the cases \(p=q=2\) obtained in [\textit{R. Latała} et al., Invent. Math. 214, No. 3, 1031--1080 (2018; Zbl 1457.60011)], and \(p=1, q\geq2\) proved in [\textit{O. Guédon} et al., Lect. Notes Math. 2169, 151--162 (2017; Zbl 1366.60010)], in the paper under review Conjecture 1 is confirmed when \(p\in \{1,\infty\}, 1\leq q \leq \infty,\) and when \(q\in \{1,\infty\}, 1\leq p \leq \infty. \) In all the other cases, the upper bounds are proved only up to logarithmic (in the dimensions \(m,n\)) multiplicative factors. Conjecture 2 holds in the same situations as Conjecture 1 is stated.
The motivation of the problems analyzed in this paper is well stated, and the presentation is very friendly for a reader even without expertise in the topic.
Reviewer: Francisco Marcellán (Leganes)Finding the area and perimeter distributions for flat Poisson processes of a straight line and Voronoi diagramshttps://zbmath.org/1540.600242024-09-13T18:40:28.020319Z"Kanel-Belov, A. Ya."https://zbmath.org/authors/?q=ai:kanel-belov.alexei"Golafshan, M."https://zbmath.org/authors/?q=ai:golafshan.m-m|golafshan.mehdi"Malev, S. G."https://zbmath.org/authors/?q=ai:malev.sergey"Yavich, R. P."https://zbmath.org/authors/?q=ai:yavich.r-pSummary: The study of distribution functions (with respect to areas, perimeters) for partitioning a plane (space) by a random field of straight lines (hyperplanes) and for obtaining Voronoi diagrams is a classical problem in statistical geometry. Moments for such distributions have been investigated since 1972 [\textit{R. E. Miles}, Adv. Appl. Probab. 4, 243--266 (1972; Zbl 0258.60015)]. We give a complete solution of these problems for the plane, as well as for Voronoi diagrams. The following problems are solved: 1. A random set of straight lines is given on the plane, all shifts are equiprobable, and the distribution law has the form \(F(\varphi )\). What is the area (perimeter) distribution of the parts of the partition? 2. A random set of points is marked on the plane. Each point \(A\) is associated with a ``region of attraction,'' which is a set of points on the plane to which \(A\) is the closest of the marked set. The idea is to interpret a random polygon as the evolution of a segment on a moving one and construct kinetic equations. It is sufficient to take into account a limited number of parameters: the covered area (perimeter), the length of the segment, and the angles at its ends. We show how to reduce these equations to the Riccati equation using the Laplace transform.Most likely balls in Banach spaces: existence and nonexistencehttps://zbmath.org/1540.600252024-09-13T18:40:28.020319Z"Schmidt, Bernd"https://zbmath.org/authors/?q=ai:schmidt.bernd-gerhard|schmidt.bernd|schmidt.bernd-hThe author establishes a general criterion for the existence of convex sets of fixed shape as, for example, balls of a given radius, of maximal probability on Banach spaces. Also, counterexamples are provided, showing that their existence may fail even in some common situations.
Reviewer: Pavel Stoynov (Sofia)Convex geometry of finite exchangeable laws and de Finetti style representation with universal correlated correctionshttps://zbmath.org/1540.600582024-09-13T18:40:28.020319Z"Carlier, Guillaume"https://zbmath.org/authors/?q=ai:carlier.guillaume"Friesecke, Gero"https://zbmath.org/authors/?q=ai:friesecke.gero"Vögler, Daniela"https://zbmath.org/authors/?q=ai:vogler.danielaSummary: We present a novel analogue for finite exchangeable sequences of the de Finetti, Hewitt and Savage theorem and investigate its implications for multi-marginal optimal transport (MMOT) and Bayesian statistics. If \((Z_1, \ldots ,Z_N)\) is a finitely exchangeable sequence of \(N\) random variables taking values in some Polish space \(X\), we show that the law \(\mu_k\) of the first \(k\) components has a representation of the form
\[
\mu_k=\int_{\mathcal{P}_{\frac{1}{N}}(X)} F_{N,k}(\lambda ) \, \text{d} \alpha (\lambda )
\]
for some probability measure \(\alpha\) on the set of \(\frac{1}{N} \)-quantized probability measures on \(X\) and certain universal polynomials \(F_{N,k} \). The latter consist of a leading term \(N^{k-1}/\prod_{j=1}^{k-1}(N - j) \lambda^{\otimes k}\) and a finite, exponentially decaying series of correlated corrections of order \(N^{-j}\) (\(j=1, \ldots ,k\)). The \(F_{N,k}(\lambda )\) are precisely the extremal such laws, expressed via an explicit polynomial formula in terms of their one-point marginals \(\lambda \). Applications include novel approximations of MMOT via polynomial convexification and the identification of the remainder which is estimated in the celebrated error bound of \textit{P. Diaconis} and \textit{D. Freedman} [Ann. Probab. 8, 745--764 (1980; Zbl 0434.60034)] between finite and infinite exchangeable laws.The mixing time of the Lozenge tiling Glauber dynamicshttps://zbmath.org/1540.602202024-09-13T18:40:28.020319Z"Laslier, Benoît"https://zbmath.org/authors/?q=ai:laslier.benoit"Toninelli, Fabio"https://zbmath.org/authors/?q=ai:toninelli.fabio-lucioSummary: The broad motivation of this work is a rigorous understanding of reversible, local Markov dynamics of interfaces, and in particular their speed of convergence to equilibrium, measured via the mixing time \(T_{mix}\). In the \((d+1)\)-dimensional setting, \(d\geqslant 2\), this is to a large extent mathematically unexplored territory, especially for discrete interfaces. On the other hand, on the basis of a mean-curvature motion heuristics [\textit{C. L. Henley}, J. Stat. Phys. 89, No. 3--4, 483--507 (1997; Zbl 0939.82029); \textit{H. Spohn}, J. Stat. Phys. 71, No. 5--6, 1081--1132 (1993; Zbl 0935.82546)] and simulations (see [\textit{N. Destainville}, ``Flip dynamics in octagonal rhombus tiling sets'', Phys. Rev. Lett. 88, Article ID 030601, 4 p. (2002; \url{doi:10.1103/PhysRevLett.88.030601})] and the references in [Henley, loc. cit.; \textit{D. B. Wilson}, Ann. Appl. Probab. 14, No. 1, 274--325 (2004; Zbl 1040.60063)]), one expects convergence to equilibrium to occur on time-scales of order \(\approx \delta^{-2}\) in any dimension, with \(\delta \rightarrow 0\) the lattice mesh.
We study the single-flip Glauber dynamics for lozenge tilings of a finite domain of the plane, viewed as \((2+1)\)-dimensional surfaces. The stationary measure is the uniform measure on admissible tilings. At equilibrium, by the limit shape theorem [\textit{H. Cohn} et al., J. Am. Math. Soc. 14, No. 2, 297--346 (2001; Zbl 1037.82016)], the height function concentrates as \(\delta \rightarrow 0\) around a deterministic profile \(\phi\), the unique minimizer of a surface tension functional. Despite some partial mathematical results [\textit{B. Laslier} and \textit{F. L. Toninelli}, Probab. Theory Relat. Fields 161, No. 3--4, 509--559 (2015; Zbl 1328.60176); Commun. Math. Phys. 338, No. 3, 1287--1326 (2015; Zbl 1329.60339); Wilson, loc. cit.], the conjecture \(T_{mix} = \delta^{-2+o(1)}\) had been proven, so far, only in the situation where \(\phi\) is an affine function [\textit{P. Caputo} et al., Commun. Math. Phys. 311, No. 1, 157--189 (2012; Zbl 1276.82034)]. In this work, we prove the conjecture under the sole assumption that the limit shape \(\phi\) contains no frozen regions (facets).Exact analytical quasibound states of a scalar particle around a Reissner-Nordström black holehttps://zbmath.org/1540.810502024-09-13T18:40:28.020319Z"Senjaya, David"https://zbmath.org/authors/?q=ai:senjaya.davidSummary: In this paper, we investigate behavior of massive and massless scalar particle around a static spherically symmetric charged black hole -- so called Reissner-Nordström black hole in \(3+1\) dimension. We successfully discover a novel exact analytical quasibound states' wave functions and energy levels by solving the covariant Klein-Gordon wave equation. The quasibound state has complex-valued energy \(E = E_R + iE_I\) where the real part \(E_R\) can be interpreted as the scalar particle's energy while the imaginary part represents the decay. We also discuss the Hawking radiation from the apparent horizon of Reissner-Nordström black hole which can be calculated using the scalar particle's wave function. In principle, this study could provide the possibility for laboratory testing of effects whose nature is absolutely related with quantum effects in gravity.A colorful Steinitz lemma with application to block-structured integer programshttps://zbmath.org/1540.901612024-09-13T18:40:28.020319Z"Oertel, Timm"https://zbmath.org/authors/?q=ai:oertel.timm"Paat, Joseph"https://zbmath.org/authors/?q=ai:paat.joseph-s"Weismantel, Robert"https://zbmath.org/authors/?q=ai:weismantel.robertSummary: The Steinitz constant in dimension \(d\) is the smallest value \(c(d)\) such that for any norm on \(\mathbb{R}^{d}\) and for any finite zero-sum sequence in the unit ball, the sequence can be permuted such that the norm of each partial sum is bounded by \(c(d)\). Grinberg and Sevastyanov prove that \(c(d) \leq d\) and that the bound of \(d\) is best possible for arbitrary norms; we refer to their result as the Steinitz Lemma. We present a variation of the Steinitz Lemma that permutes multiple sequences at one time. Our result, which we term a \textit{colorful Steinitz Lemma}, demonstrates upper bounds that are independent of the number of sequences. Many results in the theory of integer programming are proved by permuting vectors of bounded norm; this includes proximity results, Graver basis algorithms, and dynamic programs. Due to a recent paper of Eisenbrand and Weismantel, there has been a surge of research on how the Steinitz Lemma can be used to improve integer programming results. As an application we prove a proximity result for block-structured integer programs.A convex form that is not a sum of squareshttps://zbmath.org/1540.901912024-09-13T18:40:28.020319Z"Saunderson, James"https://zbmath.org/authors/?q=ai:saunderson.jamesSummary: Every convex homogeneous polynomial (or form) is nonnegative. Blekherman has shown that there exist convex forms that are not sums of squares via a nonconstructive argument. We provide an explicit example of a convex form of degree 4 in 272 variables that is not a sum of squares. The form is related to the Cauchy-Schwarz inequality over the octonions. The proof uses symmetry reduction together with the fact (due to Blekherman) that forms of even degree that are near-constant on the unit sphere are convex. Using this same connection, we obtain improved bounds on the approximation quality achieved by the basic sum-of-squares relaxation for optimizing quaternary quartic forms on the sphere.Hyperbolicity cones are amenablehttps://zbmath.org/1540.901972024-09-13T18:40:28.020319Z"Lourenço, Bruno F."https://zbmath.org/authors/?q=ai:lourenco.bruno-f"Roshchina, Vera"https://zbmath.org/authors/?q=ai:roshchina.vera"Saunderson, James"https://zbmath.org/authors/?q=ai:saunderson.jamesSummary: Amenability is a notion of facial exposedness for convex cones that is stronger than being facially dual complete (or `nice') which is, in turn, stronger than merely being facially exposed. Hyperbolicity cones are a family of algebraically structured closed convex cones that contain all spectrahedral cones (linear sections of positive semidefinite cones) as special cases. It is known that all spectrahedral cones are amenable. We establish that all hyperbolicity cones are amenable. As part of the argument, we show that any face of a hyperbolicity cone is a hyperbolicity cone. As a corollary, we show that the intersection of two hyperbolicity cones, not necessarily sharing a common relative interior point, is a hyperbolicity cone.Strong convexity of feasible sets in off-line and online optimizationhttps://zbmath.org/1540.901992024-09-13T18:40:28.020319Z"Molinaro, Marco"https://zbmath.org/authors/?q=ai:molinaro.marcoSummary: It is known that the curvature of the feasible set in convex optimization allows for algorithms with better convergence rates, and there is renewed interest in this topic for both off-line and online problems. In this paper, leveraging results on geometry and convex analysis, we further our understanding of the role of curvature in optimization:
\begin{itemize}
\item We first show the equivalence of two notions of curvature, namely, strong convexity and gauge bodies, proving a conjecture of Abernethy et al. As a consequence, this shows that the Frank-Wolfe-type method of Wang and Abernethy has accelerated convergence rate \(O\left(\frac{1}{t^2}\right)\) over strongly convex feasible sets without additional assumptions on the (convex) objective function.
\item In online linear optimization, we identify two main properties that help explaining \textit{why/when} follow the leader (FTL) has only logarithmic regret over strongly convex sets. This allows one to directly recover and slightly extend a recent result of Huang et al., and to show that FTL has logarithmic regret over strongly convex sets whenever the gain vectors are nonnegative.
\item We provide an efficient procedure for approximating convex bodies by strongly convex ones while smoothly trading off approximation error and curvature. This allows one to extend the improved algorithms over strongly convex sets to general convex sets. As a concrete application, we extend results on online linear optimization with hints to general convex sets.
\end{itemize}Strong and total duality for constrained composed optimization via a coupling conjugation schemehttps://zbmath.org/1540.902042024-09-13T18:40:28.020319Z"You, Manxue"https://zbmath.org/authors/?q=ai:you.manxue"Li, Genghua"https://zbmath.org/authors/?q=ai:li.genghuaSummary: Based on a coupling conjugation scheme and the perturbational approach, we build Fenchel-Lagrange dual problem of a composed optimization model with infinite constraints in separated locally convex spaces. This paper has mainly two targets. One is to establish strong duality under a new regularity condition \((\mathrm{RC}_A\)) and an extension closed-type condition \((\mathrm{ECRC}_A\)). The e-convex counterpart of Fenchel-Moreau theorem plays a key role in analysing the relation between them. The other aim is to achieve the sufficient and necessary characterizations for total duality in terms of \(c\)-subdifferentials. For this purpose, a formula for \(\varepsilon\)-\(c\)-subdifferentials of a proper function composed with a linear continuous operator is proved and applied.