Recent zbMATH articles in MSC 52 https://zbmath.org/atom/cc/52 2021-11-25T18:46:10.358925Z Werkzeug Polyhedral symmetry from ribbons and tubes https://zbmath.org/1472.00048 2021-11-25T18:46:10.358925Z "Boyden, Wilder" https://zbmath.org/authors/?q=ai:boyden.wilder "Farris, Frank A." https://zbmath.org/authors/?q=ai:farris.frank-a Summary: A \textit{sepak takraw} -- a ball used for a game in Thailand -- is an icosahedrally symmetric shape woven from six bands of rattan. We model it with a multi-parameter family of surfaces, all having icosahedral symmetry. This leads us to ask and answer the question: In how many other ways can we arrange some number of bands in space to create polyhedral symmetry. Our models resemble objects created by other artists; the difference here is that we use Fourier series and focus on the role of the symmetry group. Our general formulas describe a large space of potentially wonderful designs. The instructions always lead to symmetry, but perhaps bad design, until one experiments by altering the parameters. The shapes produced by this method are suitable for artistic development as digital prints or 3D sculptures. We hope that our recipes will empower readers to create their own artistic renditions. The finite matroid-based valuation conjecture is false https://zbmath.org/1472.05031 2021-11-25T18:46:10.358925Z "Tran, Ngoc Mai" https://zbmath.org/authors/?q=ai:tran.ngoc-mai Cuts in undirected graphs. I https://zbmath.org/1472.05043 2021-11-25T18:46:10.358925Z "Sharifov, F." https://zbmath.org/authors/?q=ai:sharifov.firdovsi|sharifov.f-a "Hulianytskyi, L." https://zbmath.org/authors/?q=ai:hulianytskyi.l-f|hulianytskyi.leonid Summary: This part of the paper analyzes new properties of cuts in undirected graphs, presents various models for the maximum cut problem, based on the established correspondence between the cuts in this graph and the specific bases of the extended polymatroid associated with this graph. With respect to the model, formulated as maximization of the convex function on the compact set (extended polymatroid), it was proved that the objective function has the same value at any local and global maxima, i.e., to solve the maximum cut problem, it will suffice to find a base of the extended polymatroid as a local or global maximum of the objective function. An extension of Stanley's chromatic symmetric function to binary delta-matroids https://zbmath.org/1472.05148 2021-11-25T18:46:10.358925Z "Nenasheva, M." https://zbmath.org/authors/?q=ai:nenasheva.m "Zhukov, V." https://zbmath.org/authors/?q=ai:zhukov.v-a.1|zhukov.v-o|zhukov.vadim-g|zhukov.vladimir-v|zhukov.v-d|zhukov.vitalii-vladimirovich|zhukov.v-e|zhukov.v-t|zhukov.victor-p|zhukov.vyacheslav|zhukov.v-i Summary: Stanley's symmetrized chromatic polynomial is a generalization of the ordinary chromatic polynomial to a graph invariant with values in a ring of polynomials in infinitely many variables. The ordinary chromatic polynomial is a specialization of Stanley's one. To each orientable embedded graph with a single vertex, a simple graph is associated, which is called the intersection graph of the embedded graph. As a result, we can define Stanley's symmetrized chromatic polynomial for any orientable embedded graph with a single vertex. Our goal is to extend Stanley's chromatic polynomial to embedded graphs with arbitrary number of vertices, and not necessarily orientable. In contrast to well-known extensions of, say, the Tutte polynomial from abstract to embedded graphs [\textit{C. Chun} et al., J. Comb. Theory, Ser. A 167, 7--59 (2019; Zbl 1417.05103)], our extension is based not on the structure of the underlying abstract graph and the additional information about the embedding. Instead, we consider the binary delta-matroid associated to an embedded graph and define the extended Stanley chromatic polynomial as an invariant of binary delta-matroids. We show that, similarly to Stanley's symmetrized chromatic polynomial of graphs, which satisfies 4-term relations for simple graphs, the polynomial that we introduce satisfies the 4-term relations for binary delta-matroids [\textit{S. Lando} and \textit{V. Zhukov}, Mosc. Math. J. 17, No. 4, 741--755 (2017; Zbl 1414.05067)]. For graphs, Stanley's chromatic function produces a knot invariant by means of the correspondence between simple graphs and knots. Analogously we may interpret the suggested extension as an invariant of links, using the correspondence between binary delta-matroids and links. Möbius and coboundary polynomials for matroids https://zbmath.org/1472.05162 2021-11-25T18:46:10.358925Z "Johnsen, Trygve" https://zbmath.org/authors/?q=ai:johnsen.trygve "Verdure, Hugues" https://zbmath.org/authors/?q=ai:verdure.hugues Summary: We study how some coefficients of two-variable coboundary polynomials can be derived from Betti numbers of Stanley-Reisner rings. We also explain how the connection with these Stanley-Reisner rings forces the coefficients of the two-variable coboundary polynomials and Möbius polynomials to satisfy certain universal equations. Spectral spaces of countable abelian lattice-ordered groups https://zbmath.org/1472.06024 2021-11-25T18:46:10.358925Z "Wehrung, Friedrich" https://zbmath.org/authors/?q=ai:wehrung.friedrich Summary: It is well known that the \textit{$$\ell$$-spectrum} of an Abelian $$\ell$$-group, defined as the set of all its prime $$\ell$$-ideals with the hull-kernel topology, is a completely normal generalized spectral space. We establish the following converse of this result. Theorem. Every second countable, completely normal generalized spectral space is homeomorphic to the $$\ell$$-spectrum of some Abelian $$\ell$$-group. We obtain this result by proving that a countable distributive lattice $$D$$ with zero is isomorphic to the Stone dual of some $$\ell$$-spectrum (we say that $$D$$ is \textit{$$\ell$$-representable}) iff for all $$a,b\in D$$ there are $$x,y\in D$$ such that $$a\vee b=a\vee y=b\vee x$$ and $$x\wedge y=0$$. On the other hand, we construct a non-$$\ell$$-representable bounded distributive lattice, of cardinality $$\aleph _1$$, with an $$\ell$$-representable countable $${\mathscr {L}}_{\infty ,\omega }$$-elementary sublattice. In particular, there is no characterization, of the class of all $$\ell$$-representable distributive lattices, by any class of $${\mathscr {L}}_{\infty ,\omega }$$ sentences. Extensions of Schreiber's theorem on discrete approximate subgroups in $$\mathbb{R}^d$$ https://zbmath.org/1472.11056 2021-11-25T18:46:10.358925Z "Fish, Alexander" https://zbmath.org/authors/?q=ai:fish.alexander Summary: In this paper, we give an alternative proof of Schreiber's theorem which says that an infinite discrete approximate subgroup in $$\mathbb{R}^d$$ is relatively dense around a subspace. We also deduce from Schreiber's theorem two new results. The first one says that any infinite discrete approximate subgroup in $$\mathbb{R}^d$$ is a restriction of a Meyer set to a thickening of a linear subspace in $$\mathbb{R}^d$$, and the second one provides an extension of Schreiber's theorem to the case of the Heisenberg group. Correction to: A note on Schwartz functions and modular forms'' https://zbmath.org/1472.11121 2021-11-25T18:46:10.358925Z "Rolen, Larry" https://zbmath.org/authors/?q=ai:rolen.larry "Wagner, Ian" https://zbmath.org/authors/?q=ai:wagner.ian Correction to the authors' paper [ibid. 115, No. 1, 35--51 (2020; Zbl 1470.11076)]. A higher moment formula for the Siegel-Veech transform over quotients by Hecke triangle groups https://zbmath.org/1472.11204 2021-11-25T18:46:10.358925Z "Fairchild, Samantha" https://zbmath.org/authors/?q=ai:fairchild.samantha One can begin with author's description of this research:  We compute higher moments of the Siegel-Veech transform over quotients of $$\mathrm{SL}(2, \mathbb R)$$ by the Hecke triangle groups. After fixing a normalization of the Haar measure on $$\mathrm{SL}(2, \mathbb R)$$ we use geometric results and linear algebra to create explicit integration formulas which give information about densities of $$k$$-tuples of vectors in discrete subsets of $$\mathbb R^2$$ which arise as orbits of Hecke triangle groups. This generalizes work of W. Schmidt on the variance of the Siegel transform over $$\mathrm{SL}(2, \mathbb R)/ \mathrm{SL}(2, \mathbb Z)$$.'' Such notions as the Siegel-Veech transform, the Hecke triangle group, the $$q$$-geometric Euler totient function, a translation surface, etc., are explained. A brief survey is devoted to results in the geometry of numbers, followed by background on translation surfaces, Veech groups, and the $$q$$-geometric Euler totient function''. Several auxiliary statements are proven. Also, the special attention is given to the problem how to interpret the second main theorem of this paper in terms of a counting problem. This discussion is described with explanations and figures. Toric co-Higgs sheaves https://zbmath.org/1472.14055 2021-11-25T18:46:10.358925Z "Altmann, Klaus" https://zbmath.org/authors/?q=ai:altmann.klaus "Witt, Frederik" https://zbmath.org/authors/?q=ai:witt.frederik For a complex toric variety $$X_{\Sigma}$$ given by a fan $$\Sigma\subseteq N_{\mathbb R}=N\otimes_{\mathbb Z}{\mathbb R}$$ for a lattice $$N$$ and acting torus $$T=N\otimes_{\mathbb Z}{\mathbb C}$$, by \textit{A. Klyachko} [Math. USSR, Izv. 35, No. 2, 337--375 (1990; Zbl 0706.14010)] a \textit{toric sheaf} $${\mathcal E}$$, (that is, an $${\mathcal O}_X$$-module with an action of the torus $$T$$ which is linear on the fibers and is compatible with the $$T$$-action of $$X$$) corresponds to the complex vector space $$E={\mathcal E}_1/{\mathfrak m}_{X,1}{\mathcal E}_1$$ (where $${\mathcal E}_1$$ is the stalk or $${\mathcal E}$$ at $$1\in T\subseteq X$$ and $${\mathfrak m}_{X,1}$$ is the maximal ideal of $${\mathcal O}_{X,1}$$) together with a decreasing $${\mathbb Z}$$-filtration $$E^{\bullet}_{\rho}$$ indexed by the rays $$\rho\in \Sigma(1)$$. Following \textit{I. Biswas} et al., [Ill. J. Math. 65, No. 1, 181--190 (2021; Zbl 1465.14048)] a \textit{toric co-Higgs sheaf} is a pair $$({\mathcal E},\Phi)$$ consisting of a toric sheaf $${\mathcal E}$$ on a toric variety $$X_{\Sigma}$$ and a \textit{Higgs field}, i.e., a $$T$$-equivariant $${\mathcal O}_X$$-morphism $$\Phi:{\mathcal E}\rightarrow{\mathcal E}\otimes_{{\mathcal O}_X}{\mathcal T}_X$$ such that $$\Phi\wedge\Phi=0$$, where $${\mathcal T}_X$$ is the tangent sheaf of $$X$$. Dropping the integrability condition $$\Phi\wedge\Phi=0$$, the pair $$({\mathcal E},\Phi)$$ is called a \textit{toric pre-co-Higgs sheaf} and $$\Phi$$ a \textit{pre-co-Higgs field}. The paper under review considers general co-Higgs fields and not only $$M$$-homogeneous ones as in [Biswas et al., loc. cit]. To study these general co-Higgs sheaves, the authors start characterizing pre-co-Higgs fields using Klyachko's formalism by means of associated contractions in Theorem 8 and then show that every co-Higgs field defines a commutative finitely generated $${\mathbb C}[M]$$-algebra, the \textit{Higgs algebra}. Next, the authors introduce some combinatorial invariants: First, using that a pre-co-Higgs field is a direct sum $$\Phi=\sum \Phi^r$$ of maps $$\Phi^r:{\mathcal E}\rightarrow {\mathcal E}\otimes_{{\mathcal O}_X}{\mathcal T}_X$$ of degree $$r\in M$$, they define the corresponding \textit{Higgs polytope} of $$\Phi$$ as the convex hull in $$M_{\mathbb R}$$ of its support $$\text{supp}(\Phi)=\{r\in M: \Phi^r\neq 0\}\subseteq M$$. The convex hull of the totality of degrees $$r\in M$$ of all possible toric pre-co-Higgs fields defines a second polytope, the \textit{Higgs range}. After proving some properties of these polytopes, in the last two sections of the paper they are calculated for several smooth toric surfaces: They compute the Higgs range of the projective plane and Hirzebruch and Fano surfaces, and they also compute the Higgs polytope for some del Pezzo surfaces. The whole paper includes many illustrative examples with explicit calculations nicely complementing the developments. Strong factorization and the braid arrangement fan https://zbmath.org/1472.14056 2021-11-25T18:46:10.358925Z "Machacek, John" https://zbmath.org/authors/?q=ai:machacek.john-m Summary: We establish strong factorization for pairs of smooth fans which are refined by the braid arrangement fan. Our method uses a correspondence between cones and preposets. Prism graphs in tropical plane curves https://zbmath.org/1472.14070 2021-11-25T18:46:10.358925Z "Jacoby, Liza" https://zbmath.org/authors/?q=ai:jacoby.liza "Morrison, Ralph" https://zbmath.org/authors/?q=ai:morrison.ralph "Weber, Ben" https://zbmath.org/authors/?q=ai:weber.ben Summary: Any smooth tropical plane curve contains a distinguished trivalent graph called its skeleton. In [Discrete Math. 344, No. 1, Article ID 112161, 19 p. (2021; Zbl 1455.52013)], the second author and \textit{A. K. Tewari} proved that the so-called big-face graphs cannot be the skeleta of tropical curves for genus $$12$$ and greater. In this paper we answer an open question they posed to extend their result to the prism graphs, proving that a prism graph is the skeleton of a smooth tropical plane curve precisely when the genus is at most $$11$$. Our main tool is a classification of lattice polygons with two points that can simultaneously view all others, without having any one point that can observe all others. Erratum to: Divisionally free arrangements of hyperplanes'' https://zbmath.org/1472.32014 2021-11-25T18:46:10.358925Z "Abe, Takuro" https://zbmath.org/authors/?q=ai:abe.takuro From the text: The aim of this note is to correct the statement and the proof of Theorem 6.2 in the author's paper [ibid. 204, No. 1, 317--346 (2016; Zbl 1341.32023)], which is not correct as it was stated. All the other results in the paper are correct as they were stated. On commuting billiards in higher-dimensional spaces of constant curvature https://zbmath.org/1472.37031 2021-11-25T18:46:10.358925Z "Glutsyuk, Alexey" https://zbmath.org/authors/?q=ai:glutsyuk.alexey-a Summary: We consider two nested billiards in $$\mathbb{R}^d$$, $$d\geq3$$, with $$C^2$$-smooth strictly convex boundaries. We prove that if the corresponding actions by reflections on the space of oriented lines commute, then the billiards are confocal ellipsoids. This together with the previous analogous result of the author in two dimensions solves completely the commuting billiard conjecture due to \textit{S. Tabachnikov} [Geom. Dedicata 53, No. 1, 57--68 (1994; Zbl 0813.52003)]. The main result is deduced from the classical theorem due to \textit{M. Berger} [Geometry. I, II. Transl. from the French by M. Cole and S. Levy. Berlin: Springer (2009; Zbl 1153.51001)] which says that in higher dimensions only quadrics may have caustics. We also prove versions of Berger's theorem and the main result for billiards in spaces of constant curvature (space forms). Chaotic Delone sets https://zbmath.org/1472.37047 2021-11-25T18:46:10.358925Z "López, Jesús A. Álvarez" https://zbmath.org/authors/?q=ai:alvarez-lopez.jesus-a "Lijó, Ramón Barral" https://zbmath.org/authors/?q=ai:barral-lijo.ramon "Hunton, John" https://zbmath.org/authors/?q=ai:hunton.john-robert "Nozawa, Hiraku" https://zbmath.org/authors/?q=ai:nozawa.hiraku "Parker, John R." https://zbmath.org/authors/?q=ai:parker.john-r The paper deals with chaotic Delone sets. The set $$S$$ is a Delone set in the space $$X$$ if there exists $$\varepsilon, \delta$$ such that for every $$x\in X$$ there is $$y\in S$$ with $$d(x,y)<\varepsilon$$, and $$d(x,y)\geq \delta$$ for every $$x\neq y\in S$$. There exists a natural, compact, metrizable topology on the set $$D_{\delta,\varepsilon}$$ of Delone sets which makes the following action of $$\mathbb R^n$$ continuous: $$v.S=S-v$$. Thus we have a dynamical system and the question is if this group action is chaotic. Here $$S$$ is said to be chaotic if $$S$$ is aperiodic and the union of periodic orbits is dense in the adherence of the orbit of $$S$$. Recall that a property is generic if it holds on a residual subset, i.e., a subset containing a countable intersection of open dense sets. The first theorem of the paper shows that being chaotic is a generic property if $$\varepsilon\geq \delta$$. In the second part of the paper the authors explain how to construct a chaotic Delone set in the hyperbolic plane as a cut and project set and provide its characterization. It is worth to notice that this construction can be generalized to any dimension. Equi-distributed property and spectral set conjecture on $$\mathbb{Z}_{p^2} \times \mathbb{Z}_p$$ https://zbmath.org/1472.43007 2021-11-25T18:46:10.358925Z "Shi, Ruxi" https://zbmath.org/authors/?q=ai:shi.ruxi Summary: In this paper, we show an equi-distributed property in 2-dimensional finite abelian groups $$\mathbb{Z}_{p^n} \times \mathbb{Z}_{p^m}$$, where $$p$$ is a prime number. By using this equi-distributed property, we prove that Fuglede's spectral set conjecture holds on groups $$\mathbb{Z}_{p^2} \times \mathbb{Z}_p$$, namely, a set in $$\mathbb{Z}_{p^2} \times \mathbb{Z}_p$$ is a spectral set if and only if it is a translational tile. Diametrically complete sets with empty interior in reflexive Banach spaces with the nonstrict Opial property https://zbmath.org/1472.46012 2021-11-25T18:46:10.358925Z "Kaczor, Wiesława" https://zbmath.org/authors/?q=ai:kaczor.wieslawa-j "Kuczumow, Tadeusz" https://zbmath.org/authors/?q=ai:kuczumow.tadeusz "Reich, Simeon" https://zbmath.org/authors/?q=ai:reich.simeon Summary: We prove that for each reflexive Banach space $$(X,\|\cdot\|_X)$$ with the nonstrict Opial property, there exists an equivalent norm $$\|\cdot\|_1$$ such that the Banach space $$(X,\|\cdot\|_1)$$ contains a diametrically complete set with empty interior. Schauder bases, LUR Banach spaces and diametrically with empty interior https://zbmath.org/1472.46047 2021-11-25T18:46:10.358925Z "Budzynska, Monika" https://zbmath.org/authors/?q=ai:budzynska.monika "Kaczor, Wieslawa" https://zbmath.org/authors/?q=ai:kaczor.wieslawa-j "Kot, Mariola" https://zbmath.org/authors/?q=ai:kot.mariola "Kuczumow, Tadeusz" https://zbmath.org/authors/?q=ai:kuczumow.tadeusz Summary: In this paper we prove that if a reflexive Banach space $$(X,\|\cdot\|)$$ has a Schauder basis, then it has an equivalent norm $$\|\cdot\|_0$$ such that the Banach space $$(X,\|\cdot\|)_0)$$ is LUR and has a diametrically complete set with empty interior. This result extends the Maluta theorem about existence of such a set in $$l^2$$ with the Day norm. Nonasymptotic densities for shape reconstruction https://zbmath.org/1472.49062 2021-11-25T18:46:10.358925Z "Ibrahim, Sharif" https://zbmath.org/authors/?q=ai:ibrahim.sharif "Sonnanburg, Kevin" https://zbmath.org/authors/?q=ai:sonnanburg.kevin "Asaki, Thomas J." https://zbmath.org/authors/?q=ai:asaki.thomas-j "Vixie, Kevin R." https://zbmath.org/authors/?q=ai:vixie.kevin-r Summary: In this work, we study the problem of reconstructing shapes from simple nonasymptotic densities measured only along shape boundaries. The particular density we study is also known as the integral area invariant and corresponds to the area of a disk centered on the boundary that is also inside the shape. It is easy to show uniqueness when these densities are known for all radii in a neighborhood of $$r = 0$$, but much less straightforward when we assume that we only know the area invariant and its derivatives for only one $$r > 0$$. We present variations of uniqueness results for reconstruction (modulo translation and rotation) of polygons and (a dense set of) smooth curves under certain regularity conditions. Seidel's conjectures in hyperbolic 3-space https://zbmath.org/1472.51012 2021-11-25T18:46:10.358925Z "Cussy, Omar Chavez" https://zbmath.org/authors/?q=ai:cussy.omar-chavez "Grossi, Carlos H." https://zbmath.org/authors/?q=ai:grossi.carlos-h This paper answers several conjectures raised by \textit{J. J. Seidel} [Stud. Sci. Math. Hung. 21, 243--249 (1986; Zbl 0561.52010)] for hyperbolic $$3$$-space. Given a $$4$$-dimensional $${\mathbb R}$$-linear space $$V$$ equipped with a bilinear symmetric form of signature $$---+$$, the hyperbolic $$3$$-space is the open ball of positive points $${\mathbb H}^3 = \{\mathbf{p} \in {\mathbb P}V : \langle p, p\rangle > 0\}$$, with $$p\in V$$ denoting the representative of a point $$\mathbf{p}\in {\mathbb P}V$$. If $$S := (\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3, \mathbf{v}_4)$$, with $$\mathbf{v}_i\in \partial {\mathbb H}^3$$ is an ideal tetrahedron, choosing representatives $$v_i \in V$$ we obtain a Gram matrix $$G$$ of the vertices of $$S$$, where $$G:= [\langle v_i, v_j\rangle$$. Among all the Gram matrices of the vertices of $$S$$, one, $$G_{ds}$$, is doubly stochastic. Seidel's first conjecture states that the volume of $$S$$ is determined by a preferably natural algebraic functions of the entries of $$G_{ds}$$. The authors prove more than was expected, by obtaining, using Milnor's volume formula for an ideal tetrahedron, an explicit formula for the volume of an ideal tetrahedron as a function of the permanent and the determinant of $$G_{ds}$$. While Seidel's fourth conjecture says that the volume is a decreasing function of the permanent, the authors show that this not the case, that the volume is in fact decreasing in the determinant and increasing in (the square root of) the permanent, whenever the determinant is non-zero. They also prove Seidel's third conjecture for hyperbolic $$3$$-space. Problem of shadow in the Lobachevskii space https://zbmath.org/1472.51013 2021-11-25T18:46:10.358925Z "Kostin, A. V." https://zbmath.org/authors/?q=ai:kostin.andrey-viktorovich In [\textit{G. Khudaiberganov}, On a homogeneous polynomially convex hull of the union of balls'', VINITI 21, 1772--1785 (1982)], it was proved that it is sufficient to have two disks in order to guarantee that any straight line passing through the center of a circle in the Euclidean plane crosses at least one disk centered in this circle (or, in other words, in order that the center of the circle belong to the convex 1-hull of the disks [\textit{Y. B. Zelinskii} et al., Problem of shadow and related problems,'' Proc. Int. Geom. Cent. 9, No. 3--4, 50--58 (2016; \url{doi:10.15673/tmgc.v9i3-4.319 })]). The author present the main steps of this generalization with indication of all distinctions caused by the specific features of the hyperbolic plane. This problem can be regarded as a problem of finding conditions guaranteeing that points belong to a generalized convex hull of the family of balls. A note on empty balanced tetrahedra in two-colored point sets in $$\mathbb{R}^3$$ https://zbmath.org/1472.51014 2021-11-25T18:46:10.358925Z "Díaz-Bañez, José-Miguel" https://zbmath.org/authors/?q=ai:diaz-banez.jose-miguel "Fabila-Monroy, Ruy" https://zbmath.org/authors/?q=ai:fabila-monroy.ruy "Urrutia, Jorge" https://zbmath.org/authors/?q=ai:urrutia.jorge-l Summary: Let $$S$$ be a set of $$n$$ red and $$n$$ blue points in general position in $$\mathbb{R}^3$$. Let $$\tau$$ be a tetrahedron with vertices in $$S$$. We say that $$\tau$$ is \textit{empty} if it does not contain any point of $$S$$ in its interior. We say that $$\tau$$ is \textit{balanced} if two of its vertices are blue, and two of its vertices are red. In this paper we show that $$S$$ spans $$\Omega( n^{5 / 2})$$ empty balanced tetrahedra. Realizations of the $$120$$-cell https://zbmath.org/1472.51015 2021-11-25T18:46:10.358925Z "McMullen, Peter" https://zbmath.org/authors/?q=ai:mcmullen.peter Realizations provide geometric pictures of abstract regular polytopes, and thereby help to investigate their structures. In several articles and monographs [the author, Geometric regular polytopes. Cambridge: Cambridge University Press (2020; Zbl 1454.51002); the author and \textit{E. Schulte}, Abstract regular polytopes. Cambridge: Cambridge University Press (2002; Zbl 1039.52011)] the realization spaces of all the classical regular polytops have been described except that of the 120-cell $$\{5,3,3\}$$. For several reasons, the realization domain of the 120-cell is more complicated. In this article, the author describes the realization of the 120-cell by imploying many of the techniques of the theory and the notion of cosine vectors introduced here for the first time. For the entire collection see [Zbl 1467.52001]. Arity of $$L$$-convexities in some algebraic structures https://zbmath.org/1472.52001 2021-11-25T18:46:10.358925Z "Chen, Fanhong" https://zbmath.org/authors/?q=ai:chen.fanhong "Shen, Chong" https://zbmath.org/authors/?q=ai:shen.chong Summary: This paper presents the concept of arity in $$L$$-convex spaces. It is shown that $$L$$-ordered convex spaces, standard $$L$$-convex spaces, linear $$L$$-convex spaces and affine $$L$$-convex spaces are all of arity $$\le 2$$. In addition, the convexity-preserving mappings and convex-to-convex mappings are characterized between $$L$$-convex spaces with arity property. On weak $$\epsilon$$-nets and the Radon number https://zbmath.org/1472.52002 2021-11-25T18:46:10.358925Z "Moran, Shay" https://zbmath.org/authors/?q=ai:moran.shay "Yehudayoff, Amir" https://zbmath.org/authors/?q=ai:yehudayoff.amir The main results of this paper involve a finite set $$X$$, a family $$C$$ of subsets of $$X$$ and probability measures $$\mu$$ on $$X$$. The family $$C$$ is said to have a weak $$\epsilon$$-net of size $$\beta$$ over $$\mu$$ if there exists a subset $$S$$ of $$X$$ of size $$\beta$$ whose intersection with every member of $$C$$ that has the $$\mu$$-measure at least $$\epsilon$$ is non-empty. The pair $$(X,C)$$ is called a convexity space if $$C$$ is closed under intersections and both the empty sets and the whole $$X$$ are elements of $$C$$. In this case, sets from $$C$$ are said to be convex sets of the convexity space $$(X,C)$$. The convex hull $$\mathrm{conv }Y$$ of a subset $$Y$$ of $$X$$ in the convexity space $$(X,C)$$ is defined as the intersection of all elements of $$C$$ that contain $$A$$ as a subset. A convexity space $$(X,C)$$ is called separable if for every convex set $$Y\in C$$ and a point $$p\in X\setminus Y$$ there exists a partition of $$X$$ into convex sets $$X_1\in C$$ and $$X_2\in C$$ that satisfies $$Y\subseteq X_1$$ and $$p\in X_2$$. The Radon number of a convexity space $$(X,C)$$ is the minimal number $$r$$ with the property that every subset $$Y$$ of $$X$$ with $$r$$ elements can be partitioned into sets $$Y_1$$ and $$Y_2$$ whose convex hulls have a non-empty intersection. The main theorem of this paper is about the interplay of the Radon number and the size of weak $$\epsilon$$-nets for $$X$$. The theorem states that for a finite convexity space $$(X,C)$$ the following assertions hold: \begin{itemize} \item[(a)] if $$(X,C)$$ is separable and has Radon number at most $$r$$, then, for every $$\epsilon>0$$ and every probability measure $$\mu$$ on $$X$$, the family $$C$$ has a weak $$\epsilon$$-net over $$\mu$$ of size at most $$(120 r^2/\epsilon) ^{4 r^2\ln(1/\epsilon)}$$ \item[(b)] If $$(X,C)$$ has Radon number at least $$r$$, then for some choice of a probability measure $$\mu$$ on $$X$$, every 1/4-net for $$C$$ over $$\mu$$ has size at least $$r/2$$. \end{itemize} The authors mention that meanwhile Holmsen and Lee (2019) have derived an upper bound on the size of the weak $$\epsilon$$-net of finite convexity spaces $$(X,C)$$ without any use of the separability assumption on $$(X,C)$$. $$\mathbb{B}$$-spaces are KKM spaces https://zbmath.org/1472.52003 2021-11-25T18:46:10.358925Z "Park, Sehie" https://zbmath.org/authors/?q=ai:park.sehie Summary: A subset $$B$$ of $$\mathbb{R}_+^n$$ is $$\mathbb{B}$$-convex if for all $$x_1,x_2\in B$$ and all $$t\in[0,1]$$ one has $$tx_1\vee x_2\in B$$. These sets were first investigated in [\textit{W. Briec} and \textit{C. Horvath}, Optimization 53, No. 2, 103--127 (2004; Zbl 1144.90506)]. In this paper, we show that any finite dimensional $$\mathbb{B}$$-space is a KKM space, that is, a space satisfying the abstract form of the celebrated Knaster-Kuratowski-Mazurkiewicz theorem appeared in 1929 and its open-valued version. Therefore, a $$\mathbb{B}$$-space satisfies a large number of the KKM theoretic results appeared in the literature. Dimension of the space of unitary equivariant translation invariant tensor valuations https://zbmath.org/1472.52004 2021-11-25T18:46:10.358925Z "Böröczky, K. J." https://zbmath.org/authors/?q=ai:boroczky.karoly-jun "Domokos, M." https://zbmath.org/authors/?q=ai:domokos.matyas "Solanes, G." https://zbmath.org/authors/?q=ai:solanes.gil Suppose that $${\mathbb V}, {\mathbb W}$$ are finite-dimensional real vector spaces and $$Z$$ is a translation-invariant, continuous valuation on the space $${\mathcal K}({\mathbb V})$$ of convex bodies in $${\mathbb V}$$ (equipped with the Hausdorff metric) with values in $${\mathbb W}$$. If $$G\subset\mathrm{GL}({\mathbb V})$$ is a closed subgroup acting on $${\mathbb W}$$, the valuation $$Z$$ is called $$G$$-equivariant if $$Z(\varphi(K))=\varphi Z(K)$$ for all $$\varphi\in G$$ and $$K\in {\mathcal K}({\mathbb V})$$. In the present paper, $${\mathbb V}={\mathbb R}^{2m}$$ (where $$m\ge 2$$), which can be identified with $${\mathbb C}^m$$, further $$G=\mathrm{U}(m)$$ is the unitary group, and $${\mathbb W}$$ is the space $${\mathbb S}^d({\mathbb R}^{2m})$$ of symmetric tensors of rank $$d$$ on $${\mathbb V}$$. With these choices, the paper determines the (finite) dimensions of the spaces of $$k$$-homogeneous, translation-invariant, continuous, $$\mathrm{U}(m)$$-equivariant valuations. The dimensions are expressed explicitly in terms of $$m,d,k$$. For $$d=0$$, the result is due to \textit{S. Alesker} [J. Differ. Geom. 63, No. 1, 63--95 (2003; Zbl 1073.52004)], and for $$d=1$$ to \textit{T. Wannerer} [J. Differ. Geom. 96, No. 1, 141--182 (2014; Zbl 1296.53149)]. The proofs make heavy use of representation theory, and also Wannerer's [loc. cit.] theory of unitarily invariant area measures proves useful. Leaves decompositions in Euclidean spaces https://zbmath.org/1472.52005 2021-11-25T18:46:10.358925Z "Ciosmak, Krzysztof J." https://zbmath.org/authors/?q=ai:ciosmak.krzysztof-j Summary: We partly extend the localisation technique from convex geometry to the multiple constraints setting. For a given 1-Lipschitz map $$u:\mathbb{R}^n\to\mathbb{R}^m$$, $$m\leq n$$, we define and prove the existence of a partition of $$\mathbb{R}^n$$, up to a set of Lebesgue measure zero, into maximal closed convex sets such that restriction of $$u$$ is an isometry on these sets. We consider a disintegration, with respect to this partition, of a log-concave measure. We prove that for almost every set of the partition of dimension $$m$$, the associated conditional measure is log-concave. This result is proven also in the context of the curvature-dimension condition for weighted Riemannian manifolds. This partially confirms a conjecture of Klartag. Mixed complex brightness integrals https://zbmath.org/1472.52006 2021-11-25T18:46:10.358925Z "Li, Chao" https://zbmath.org/authors/?q=ai:li.chao.5|li.chao.2|li.chao.1 "Wang, Weidong" https://zbmath.org/authors/?q=ai:wang.weidong "Lin, Youjiang" https://zbmath.org/authors/?q=ai:lin.youjiang The authors introduce a notion of mixed brightness of complex convex bodies (convex and compact subsets of $$\mathbb{C}^n$$), and prove several inequalities for these quantities. More precisely, following \textit{J. Abardia} [J. Funct. Anal. 263, No. 11, 3588--3603 (2012; Zbl 1262.52012)], given a convex body $$M$$ in $$\mathbb{C}^n$$ and a convex subset $$C$$ of $$\mathbb{C}$$, the authors consider the brightness function of $$M$$, defined as follows: $\delta^C(M,\omega)=\frac12 V(M[2n-1],\omega),\quad\omega\in S^{2n-1}.$ Here $$S^{2n-1}$$ is the unit sphere in $$\mathbb{C}^n$$, and $$V$$ denotes the mixed volume operator. The mixed complex brightness integral of $$2n$$ convex bodies $$M_1,\dots,M_{2n}$$, is then defined as follows: $D^C(M_1,\dots,M_{2n})=\frac1{2n}\int_{S^{2n-1}}\delta^C(M_1,\omega)\cdots\delta^C(M_{2n},\omega)dS(\omega),\tag{1}$ where $$dS$$ denotes the integration with respect to the $$(2n-1)$$-dimensional Hausdorff measure restricted to $$S^{2n-1}$$. The main result of the paper is the following family of inequalities. Theorem. Let $$m\in\{2,\dots,2n\}$$, $$M_1,\dots,M_{2n}$$ be convex bodies in $${\mathbb C}^n$$, and $$C$$ be a convex subset of $$\mathbb C$$. Then $D^C(M_1,\dots,M_{2n})^m\le\prod_{i=1}^m D^C(M_1,\dots,M_{2n-m},\underbrace{M_{2n-i+1},\dots,M_{2n-i+1}}_{m}).$ A complete characterization of equality conditions is also established. A second family of inequalities is proved. Theorem. Let $$M,N$$ be convex bodies in $${\mathbb C}^n$$, let $$C\subset{\mathbb C}$$ be a convex set, and let $$i,j,k$$ be such that $$i<j<k$$. Then $D_j^C(M,N)^{k-i}\le D_i^C(M,N)^{k-j}D_k^C(M,N)^{j-i}.$ Here, for a general index $$i\in\{0,\dots,2n\}$$, $D_i^C(M,N)=D^C(\underbrace{M,\dots,M}_{2n-i},\underbrace{N,\dots,N}_{i})$ (this notion can be extended to arbitrary real values of $$i$$, using an integral expression of type (1)). Several further inequalities for mixed complex brightness integrals are established as corollaries of the previous two theorems (for instance isoperimetric type inequalities). Penumbras and separation of convex sets https://zbmath.org/1472.52007 2021-11-25T18:46:10.358925Z "Soltan, Valeriu" https://zbmath.org/authors/?q=ai:soltan.valeriu The penumbra $$P(K_1,K_2)$$ of two convex $$K_1$$ and $$K_2$$ in $$\mathbb{R}^n$$ is the set of all points of the form $$x_1+\alpha(x_2-x_1)$$ with $$x_1\in K_1$$, $$x_2\in K_2$$ and $$\alpha\ge 0$$. In other words, $$P(K_1, K_2)$$ can be described as $$K_1+\mathrm{cone}(K_2 -K_1)$$, where $$\mathrm{cone}$$ is the conic hull operation (this is mentioned in Theorem 10). The paper contains 13 theorems on intrinsic convex-geometric properties of penumbras. In particular, Theorems 1 and 2 are about properties related to common convex-geometric operations and functionals, including affine hull, relative interior, closure and the Hausdorff distance; Theorem 3 characterizes different versions of separation of $$K_1$$ and $$K_2$$ via separation of the respective penumbras $$P(K_1,K_2)$$ and $$P(K_2,K_1)$$; Theorem 4 deals with supporting hyperplanes of the topological closure of a penumbra and Theorem 5 describes the closure $$P(K_1,K_2)$$ in terms of half-spaces that separate $$K_1$$ from $$K_2$$. Asymptotic estimates for the largest volume ratio of a convex body https://zbmath.org/1472.52008 2021-11-25T18:46:10.358925Z "Galicer, Daniel" https://zbmath.org/authors/?q=ai:galicer.daniel "Merzbacher, Mariano" https://zbmath.org/authors/?q=ai:merzbacher.mariano "Pinasco, Damián" https://zbmath.org/authors/?q=ai:pinasco.damian Summary: The \textit{largest volume ratio} of a given convex body $$K \subset \mathbb{R}^n$$ is defined as $\operatorname{lvr}(K):= \sup\limits_{L \subset \mathbb{R}^n} \operatorname{vr}(K,L),$ where the sup runs over all the convex bodies $$L$$. We prove the following sharp lower bound: $c \sqrt{n} \leq \operatorname{lvr}(K),$ for \textit{every} body $$K$$ (where $$c > 0$$ is an absolute constant). This result improves the former best known lower bound, of order $$\sqrt{{n}/{\log \log(n)}}$$. We also study the exact asymptotic behaviour of the largest volume ratio for some natural classes. In particular, we show that $$\text{lvr}(K)$$ behaves as the square root of the dimension of the ambient space in the following cases: if $$K$$ is the unit ball of an unitary invariant norm in $$\mathbb{R}^{d \times d}$$ (e.g., the unit ball of the $$p$$-Schatten class $$S_p^d$$ for any $$1 \leq p \leq \infty$$), if $$K$$ is the unit ball of the full/symmetric tensor product of $$\ell_p$$-spaces endowed with the projective or injective norm, or if $$K$$ is unconditional. Fine approximation of convex bodies by polytopes https://zbmath.org/1472.52009 2021-11-25T18:46:10.358925Z "Naszódi, Márton" https://zbmath.org/authors/?q=ai:naszodi.marton "Nazarov, Fedor" https://zbmath.org/authors/?q=ai:nazarov.fedor-l "Ryabogin, Dmitry" https://zbmath.org/authors/?q=ai:ryabogin.dmitry For every convex body $$K$$ in $$d$$-dimensional Euclidean space with the center of mass at the origin and every $$0<\varepsilon<1/2$$, it is shown that there exists a convex polytope $$P$$ having at most $$e^{O(d)} \varepsilon^{-\frac{d-1}2}$$ vertices satisfying $$(1-\varepsilon) K\subset P\subset K$$, i.e. $$\varepsilon$$-approximating $$P$$ w.r.t. Banach-Mazur distance. This improves the result of [\textit{A. Barvinok}, Int. Math. Res. Not. 2014, No. 16, 4341--4356 (2014; Zbl 1300.52007)] by removing an extra factor of $$(\log\frac1\varepsilon)^d$$ and not requiring symmetry. Geometric and probabilistic methods are used, including the Blaschke-Santaló inequality and its reverse. Strong independence and the dimension of a Tverberg set https://zbmath.org/1472.52010 2021-11-25T18:46:10.358925Z "Choudhury, Snigdha Bharati" https://zbmath.org/authors/?q=ai:choudhury.snigdha-bharati "Deo, Satya" https://zbmath.org/authors/?q=ai:deo.satya.1|deo.satya.2 A set $$S$$ in $$\mathbb{R}^d$$ is said to be: \begin{itemize} \item[(i)] in general position, if no $$q$$ points of $$S$$ lie in a $$(q - 2)$$-dimensional flat, for $$q = 2, 3, \dots , d + 1$$; \item[(ii)] strongly independent, if for each finite family $$\{S_1, \dots, S_r\}$$ of pairwise disjoint subsets of $$S$$, with $$\max (\operatorname{card} S_i) \leq d +1$$, $$\dim \big( \bigcap_{i = 1}^r \operatorname{aff} S_i\big) = \max \big(-1, d - \sum_{i = 1}^r (d + 1 - \operatorname{card} S_i)\big)$$; \item[(iii)] weakly independent provided $$S$$ is in general position, and no subspaces of positive dimension spanned by vertex-disjoint subsets of $$S$$ are parallel; \item[(iv)] algebraically independent if the set of all real coordinates of the points of $$S$$ is algebraically independent over $$\mathbb{Q}$$. \end{itemize} In this paper, the authors establish several relationships between the four concepts mentioned above. Further consequences of the colorful Helly hypothesis https://zbmath.org/1472.52011 2021-11-25T18:46:10.358925Z "Martínez-Sandoval, Leonardo" https://zbmath.org/authors/?q=ai:martinez-sandoval.leonardo "Roldán-Pensado, Edgardo" https://zbmath.org/authors/?q=ai:roldan-pensado.edgardo "Rubin, Natan" https://zbmath.org/authors/?q=ai:rubin.natan Summary: Let $$\mathcal{F}$$ be a family of convex sets in $$\mathbb{R}^d,$$ which are colored with $$d + 1$$ colors. We say that $$\mathcal{F}$$ satisfies the Colorful Helly Property if every rainbow selection of $$d + 1$$ sets, one set from each color class, has a non-empty common intersection. The Colorful Helly Theorem of Lovász states that for any such colorful family $$\mathcal{F}$$ there is a color class $$\mathcal{F}_i \subset \mathcal{F},$$ for $$1 \le i \le d + 1,$$ whose sets have a non-empty intersection. We establish further consequences of the Colorful Helly hypothesis. In particular, we show that for each dimension $$d \ge 2$$ there exist numbers $$f(d)$$ and $$g(d)$$ with the following property: either one can find an additional color class whose sets can be pierced by $$f(d)$$ points, or all the sets in $$\mathcal{F}$$ can be crossed by $$g(d)$$ lines. Extremal convex bodies for affine measures of symmetry https://zbmath.org/1472.52012 2021-11-25T18:46:10.358925Z "Safronenko, Evgenii" https://zbmath.org/authors/?q=ai:safronenko.evgenii \textit{M. Meyer} et al. introduced in [Adv. Math. 228, No. 5, 2920--2942 (2011; Zbl 1241.52003)] affine measures of symmetry for a convex body $$K$$ based on a pair $$(p_1(K),p_2(K))$$ of affine covariant points in $$K$$, e.g., the centroid $$g(K)$$, the Santaló point $$s(K)$$ or the John point $$j(K)$$ and Löwner point $$l(K)$$, which are the centers of the John and Löwner ellipsoid, respectively. In the paper under review the author gives bounds for this measure in the cases $$(g(K),j(K))$$, $$(g(K),l(K))$$. For the pair $$(j(K),l(K))$$ see also [\textit{O. Mordhorst}, Isr. J. Math. 219, No. 2, 529--548 (2017; Zbl 1376.52015)]. Parity representations of posets https://zbmath.org/1472.52013 2021-11-25T18:46:10.358925Z "Lawrence, Jim" https://zbmath.org/authors/?q=ai:lawrence.james A \textit{parity representation} of a ranked poset $$P$$ is a mapping $$f:P\to \mathbb Z^{d }$$ that maps each chain of $$P$$ to the vertex set of a unimodular simplex in such a way that the collection of simplexes so obtained forms a geometric cell complex in $$\mathbb R^{d}$$, with $$f$$ also satisfying the conditions \begin{enumerate} \item[(i)] if $$x,y\in P$$ and $$x\le y$$, then the set of indices of odd components of $$f(x)$$ is a subset of that of $$y$$, and \item[(ii)] the rank of $$x\in P$$ equals the number of odd components of $$f(x)$$. \end{enumerate} We denote the poset of all intervals of a poset $$P$$ and the poset of all nonempty intervals of $$P$$ by $$\mathcal I(P)$$ and $$\mathcal J(P)$$, respectively. The main result of the paper is that, if $$P$$ is a ranked poset with given parity representation, then there is a canonically associated parity representation of the ranked poset $$\mathcal J(P)$$; see Theorem 4.1. There is also an analogous result for the poset $$\mathcal I(P)$$; see Theorem 4.3. Given a finite (ranked) poset $$P$$ and a nonnegative integer $$k$$, we denote by $$\mathcal J^{k}(P)$$ the result of $$k$$ iterations of $$\mathcal J$$; here $$\mathcal J^{0}(P)$$ is $$P$$. Enumeration of the elements of $$\mathcal J^{k}(P)$$ and connections to Ehrhart polynomials are also discussed in the paper (Theorem 5.1). The general formula for the Ehrhart polynomial of polytopes with applications https://zbmath.org/1472.52014 2021-11-25T18:46:10.358925Z "Sadiq, Fatema A." https://zbmath.org/authors/?q=ai:sadiq.fatema-a "Salman, Shatha A." https://zbmath.org/authors/?q=ai:salman.shatha-a "Sabri, Raghad I." https://zbmath.org/authors/?q=ai:sabri.raghad-i Summary: Recently, polytopes have shown wide applications in a lot of situations. For example, a cyclic polytope is very important in different areas of science like solutions to extremum problems (the Upper Bound Conjecture). Polytopes serve as bases for diverse constructions (from triangulations to bimatrix games). In addition, we give the general form for the product of simplex polytopes and an algorithm for these computations. Locally toroidal polytopes of rank 6 and sporadic groups https://zbmath.org/1472.52015 2021-11-25T18:46:10.358925Z "Pasechnik, Dmitrii V." https://zbmath.org/authors/?q=ai:pasechnik.dmitrii-v Summary: We augment the list of finite universal locally toroidal regular polytopes of type $$\{3, 3, 4, 3, 3 \}$$ due to \textit{P. McMullen} and \textit{E. Schulte} [Abstract regular polytopes. Cambridge: Cambridge University Press (2002; Zbl 1039.52011)], adding as well as removing entries. This disproves a related long-standing conjecture. Our new universal polytope is related to a well-known $$Y$$-shaped presentation for the sporadic simple group $$\mathit{Fi}_{22}$$, and admits $$S_4 \times O_8^+(2) : S_3$$ as the automorphism group. We also discuss further extensions of its quotients in the context of $$Y$$-shaped presentations. As well, we note that two known examples of finite universal polytopes of type $$\{3, 3, 4, 3, 3 \}$$ are related to $$Y$$-shaped presentations of orthogonal groups over $$\mathbb{F}_2$$. Mixing construction is used in a number of places to describe covers and 2-covers. The equivariant Ehrhart theory of the permutahedron https://zbmath.org/1472.52016 2021-11-25T18:46:10.358925Z "Ardila, Federico" https://zbmath.org/authors/?q=ai:ardila.federico "Supina, Mariel" https://zbmath.org/authors/?q=ai:supina.mariel "Vindas-Meléndez, Andrés R." https://zbmath.org/authors/?q=ai:vindas-melendez.andres-r In [\textit{A. Stapledon}, Adv. Math. 226, No. 4, 3622--3654 (2011; zbl 1218.52014)] Stapledon introduced \textit{equivariant Ehrhart theory}, a variant of Ehrhart theory that takes group actions into account. For a lattice polytope $$P$$ whose vertices lie in the lattice $$M$$ and a group $$G$$ acting on $$M$$, one can define the \textit{equivariant $$H^*$$-series} $$H^*[z]$$ which can be written as $$\sum_{i\geq 0} H_i^*z^i$$ for appropriate virtual characters $$H_i^*$$. Stapledon asks whether or not this series is effective, i.e, whether all the $$H_i^*$$ are characters of representations of $$G$$, and proposes the \textit{effectiveness conjecture} which states that the effectiveness of the equivariant $$H^*$$-series is equivalent to two other properties, namely \begin{itemize} \item[(i)] the toric variety of $$P$$ admits a $$G$$-invariant non-degenerate hypersurface, \item[(ii)] the equivariant $$H^*$$-series is a polynomial. \end{itemize} It is already known that (i) is a sufficient and (ii) is a necessary condition. The present paper proves the effectiveness conjecture and three minor conjectures in the case of permutahedra under the action of the symmetric group. The complete classification of empty lattice 4-simplices https://zbmath.org/1472.52017 2021-11-25T18:46:10.358925Z "Iglesias-Valiño, Óscar" https://zbmath.org/authors/?q=ai:iglesias-valino.oscar "Santos, Francisco" https://zbmath.org/authors/?q=ai:santos.francisco Summary: An empty simplex is a lattice simplex with only its vertices as lattice points. Their classification in dimension three was completed by G. White in 1964. In 1988, S. Mori, D. R. Morrison, and I. Morrison started the task in dimension four, with their motivation coming from the close relationship between empty simplices and terminal quotient singularities. They conjectured a classification of empty simplices of prime volume, modulo finitely many exceptions. Their conjecture was proved by Sankaran (1990) with a simplified proof by Bober (2009). The same classification was claimed by Barile et al. in 2011 for simplices of non-prime volume, but this statement was proved wrong by Blanco et al. (2016). In this article, we complete the classification of 4-dimensional empty simplices. In doing so, we correct and complete the classification by Barile et al., and we also compute all the finitely many exceptions, by first proving an upper bound for their volume. The whole classification has: \begin{itemize} \item[1)] One 3-parameter family, consisting of simplices of width equal to one. \item[2)] Two 2-parameter families (the one in Mori et al., plus a second new one). \item[3)] Forty-six 1-parameter families (the 29 in Mori et al., plus 17 new ones). \item[4)] 2461 individual simplices not belonging to the above families, with (normalized) volumes ranging between 24 and 419. \end{itemize} We characterize the infinite families of empty simplices in terms of the lower dimensional point configurations that they project to, with techniques that can potentially be applied to higher dimensions and other classes of lattice polytopes. Random Gale diagrams and neighborly polytopes in high dimensions https://zbmath.org/1472.52018 2021-11-25T18:46:10.358925Z "Schneider, Rolf" https://zbmath.org/authors/?q=ai:schneider.rolf-g A convex polytope $$P$$ in Euclidean space $$\mathbb{R}^d$$ is $$k$$-neighborly if any $$k$$ or fewer vertices of $$P$$ are neighbors, i.e. if their convex hull is a face of $$P$$. The author recalls a suggestion of David Gale from 1956 and generates sets of combinatorially isomorphic polytopes by choosing their Gale diagrams at random. Importantly, the paper provides a definition of a random Gale diagram. Inspired by a result of [\textit{D. L. Donoho} and \textit{J. Tanner}, Proc. Natl. Acad. Sci. USA 102, No. 27, 9452--9457 (2005; Zbl 1135.60300)], Theorem 1 shows that in high dimensions and under suitable assumptions on the growth of several parameters, the obtained random polytopes have strong neighborliness properties with high probability. Theorem 2 considers the expectation of the involved random variables and describes a phase transition with an explicit threshold. Multi-splits and tropical linear spaces from nested matroids https://zbmath.org/1472.52019 2021-11-25T18:46:10.358925Z "Schröter, Benjamin" https://zbmath.org/authors/?q=ai:schroter.benjamin Summary: We present an explicit combinatorial description of a special class of facets of the secondary polytopes of hypersimplices. These facets correspond to polytopal subdivisions called multi-splits. We show that the maximal cells in a multi-split of a hypersimplex are matroid polytopes of nested matroids. Moreover, we derive a description of all multi-splits of a product of simplices. Additionally, we present a computational result to derive explicit lower bounds on the number of facets of secondary polytopes of hypersimplices. Continuous flattening of all polyhedral manifolds using countably infinite creases https://zbmath.org/1472.52020 2021-11-25T18:46:10.358925Z "Abel, Zachary" https://zbmath.org/authors/?q=ai:abel.zachary-r "Demaine, Erik D." https://zbmath.org/authors/?q=ai:demaine.erik-d "Demaine, Martin L." https://zbmath.org/authors/?q=ai:demaine.martin-l "Ku, Jason S." https://zbmath.org/authors/?q=ai:ku.jason-s "Lynch, Jayson" https://zbmath.org/authors/?q=ai:lynch.jayson "Itoh, Jin-ichi" https://zbmath.org/authors/?q=ai:itoh.jin-ichi "Nara, Chie" https://zbmath.org/authors/?q=ai:nara.chie Summary: We prove that any finite polyhedral manifold in 3D can be continuously flattened into 2D while preserving intrinsic distances and avoiding crossings, answering a 19-year-old open problem, if we extend standard folding models to allow for countably infinite creases. The most general cases previously known to be continuously flattenable were convex polyhedra and semi-orthogonal polyhedra. For non-orientable manifolds, even the existence of an instantaneous flattening (flat folded state) is a new result. Our solution extends a method for flattening semi-orthogonal polyhedra: slice the polyhedron along parallel planes and flatten the polyhedral strips between consecutive planes. We adapt this approach to arbitrary nonconvex polyhedra by generalizing strip flattening to nonorthogonal corners and slicing along a countably infinite number of parallel planes, with slices densely approaching every vertex of the manifold. We also show that the area of the polyhedron that needs to support moving creases (which are necessary for closed polyhedra by the Bellows Theorem) can be made arbitrarily small. The minimum number of interior $$H$$-points of convex $$H$$-dodecagons https://zbmath.org/1472.52021 2021-11-25T18:46:10.358925Z "Wei, X." https://zbmath.org/authors/?q=ai:wei.xinjie|wei.xu|wei.xianzhu|wei.xuemei|wei.xiujuan|wei.xiaoqiao|wei.xi|wei.xuguang|wei.xianbiao|wei.xiaoliang|wei.xiaoqi|wei.xue|wei.xuehui|wei.xiaoyan|wei.xiulei|wei.xizhang|wei.xiaohui|wei.ximei|wei.xingyu|wei.xiyuan|wei.xiaodong|wei.xinzhu|wei.xiaoqian|wei.xinguo|wei.xuexia|wei.xiaodi|wei.xiaofan|wei.xiaoping|wei.xiuqin|wei.xiaofeng|wei.xilei|wei.xiaofei|wei.xueyong|wei.xiaoxi|wei.xing|wei.xuting|wei.xuehong|wei.xilin|wei.xianwu|wei.xianhua|wei.xinyuan|wei.xiaochao|wei.xian|wei.xiong|wei.xiujie|wei.xiangdong|wei.xia|wei.xianglan|wei.xiuxi|wei.xiangfei|wei.xizhu|wei.xueye|wei.xingmin|wei.xin|wei.xiaohai|wei.xiaoying|wei.xiaobin|wei.xuan|wei.xiaojing|wei.xiaogang|wei.xinquan|wei.xianyong|wei.xiaochen|wei.xianpeng|wei.xiujin|wei.xifeng|wei.xiaowen|wei.xueying|wei.xiansun|wei.xianglin|wei.xiaohao|wei.xiaomin|wei.xiaona|wei.xiangni|wei.xiangzhi|wei.xueliang|wei.xiao|wei.xiaopeng|wei.xiaohan|wei.xiaomei|wei.xuefeng|wei.xuewu|wei.xiaoxiao|wei.xiaowei|wei.xiaodan|wei.xiangqing|wei.xile|wei.xinyu|wei.xiaojuan|wei.xueqi|wei.xuetao|wei.xiukun|wei.xuerun|wei.xiaoling|wei.xiuling|wei.xinjiang|wei.xuerui|wei.xusheng|wei.xiaojin|wei.xinxing|wei.xiang|wei.xiangyu|wei.xiaoyu|wei.xuegang|wei.xiaoyao|wei.xiaoran|wei.xuqing|wei.xiaoli "Wang, W." https://zbmath.org/authors/?q=ai:wang.wenge|wang.wilson|wang.weiyin|wang.weiyu|wang.wuliang|wang.wutian|wang.weitong|wang.wenying|wang.weilue|wang.wanru|wang.weibiao|wang.wenching|wang.wei.30|wang.weimin|wang.wenrong|wang.weilian|wang.wenrui|wang.wenke|wang.weidong|wang.weikun|wang.weiyue|wang.wanjie|wang.wenchao|wang.wenfa|wang.wenlong|wang.wenzheng|wang.wuling|wang.weichen|wang.weibing|wang.wenqing|wang.wenjun|wang.wenjia|wang.wanlan|wang.wuhong|wang.weihsin|wang.wensheng.2|wang.wanzhong|wang.weiyang|wang.wendy|wang.weiben|wang.wenkai|wang.wencong|wang.wenjin|wang.wenyang|wang.wenzhi|wang.weiwei|wang.wenguang|wang.weijia|wang.weiqun|wang.wanneng|wang.weiguo|wang.wenzhong|wang.weijuan|wang.weina|wang.weijing|wang.weirong|wang.wenshuai|wang.weiren|wang.wenhao|wang.weihan|wang.wencheng|wang.wenkang|wang.wensheng.1|wang.weihong|wang.wenwu|wang.wenpei|wang.weining|wang.wu|wang.weisheng|wang.wenjing|wang.weihui|wang.weiqi|wang.wanxiong|wang.wan-ying|wang.wei-fan|wang.wang|wang.wenxue|wang.weibin|wang.wentong|wang.wen|wang.whedy|wang.wenwei|wang.wenqi|wang.wenbo|wang.wenyou|wang.weiguang|wang.wenxia|wang.wenbin|wang.wenfei|wang.wenhe|wang.wengege|wang.weize|wang.weiliang|wang.weiqung|wang.wenya|wang.wusheng|wang.wengdi|wang.weiye|wang.weichi|wang.weixiang|wang.wenqiang|wang.wenwen|wang.wubao|wang.weihua.1|wang.wanli|wang.wenqin|wang.weizhen|wang.wenxu|wang.wanpeng|wang.weishu|wang.wumin|wang.wenbin.1|wang.wenliang|wang.wenkui|wang.wenshu|wang.weiyao|wang.weiqin|wang.weinong|wang.wenhuan|wang.wanbin|wang.wenxuan|wang.wenna|wang.wenguan|wang.wie|wang.wenhui|wang.wengqia|wang.wenjian|wang.we|wang.wendun|wang.weijin|wang.wenzhou|wang.wanheng|wang.wenming|wang.wenping|wang.wenyu|wang.weiqian|wang.wenqian|wang.wuyang|wang.weixuan|wang.wenmin|wang.wenlin|wang.wenquan|wang.wanshan|wang.wenye|wang.weijun|wang.weiwu|wang.wenzhao|wang.weiqiang.2|wang.wenshan|wang.wendong|wang.weiran|wang.wenyan|wang.weizhe|wang.wenxiang|wang.weiying|wang.wenhu|wang.wencan|wang.wenbing|wang.weichao|wang.weizheng|wang.wannan|wang.weichung|wang.weizhong|wang.wenyong|wang.wenju|wang.weilong|wang.weihao|wang.weiyi|wang.wenchuan|wang.weiling|wang.wenfu|wang.weili|wang.wenyuan|wang.wanncherng|wang.weixing|wang.weicang|wang.wansen|wang.weigang|wang.wendan|wang.wenxi|wang.weixin|wang.wensheng|wang.wanwan|wang.wanjun|wang.weizhi|wang.wanyu|wang.wenshuang|wang.wanyong|wang.weixian|wang.weinju|wang.wenpeng|wang.weitao|wang.wendi|wang.weinan|wang.wenqia|wang.wanyi|wang.wei-min|wang.wennai|wang.wenxiu|wang.weiping.1|wang.weilan|wang.weini|wang.wentao|wang.weifeng|wang.weijie|wang.weishen|wang.wenli|wang.wanliang|wang.wenfeng|wang.wuyi|wang.wenling|wang.weihua|wang.wenbiao|wang.weihu|wang.weiqiong|wang.wuqian|wang.weixiong|wang.wanxin|wang.weizhao|wang.weilin|wang.wenxian|wang.wenxin|wang.weijiang|wang.wenhua|wang.weiming|wang.wenhai|wang.wenjie|wang.wansheng|wang.weizhuo|wang.weiping|wang.wenyi|wang.weiqing|wang.wan|wang.weibo|wang.wenjuan|wang.wenzhe|wang.weiqiang.1|wang.wanting|wang.weijian|wang.weixia|wang.weike|wang.wenchang|wang.wensong|wang.wugang|wang.wudi|wang.weiru|wang.wenxing|wang.weifang|wang.wenting "Guo, Z." https://zbmath.org/authors/?q=ai:guo.zheng|guo.zhirong|guo.zijie|guo.zhijia|guo.zhaoxia|guo.zonghe|guo.zihao|guo.zongyi|guo.zhengfeng|guo.zixi|guo.zhengru|guo.zonghuai|guo.zhenhua|guo.zunguang|guo.zhixiong|guo.zixiong|guo.zhenyi|guo.zhonglin|guo.zhiping|guo.zhenxi|guo.zhanhai|guo.zhenghong|guo.zhilin|guo.zhurui|guo.zhichuan|guo.zhiqiang|guo.zhongming|guo.zhaohui|guo.zhiyi|guo.zhongheng|guo.zhenxiong|guo.zhenkai|guo.zhenyu|guo.zihai|guo.zhaopu|guo.zhongze|guo.zong-kuan|guo.zhongyin|guo.zhenchao|guo.zihua|guo.zhiheng|guo.zilong|guo.zijian|guo.zhishan|guo.zhankuan|guo.zhixue|guo.zhen|guo.zuchao|guo.zizheng|guo.zhonghai|guo.zhenbo|guo.zihuan|guo.zhaozhuang|guo.zhi|guo.zhihua|guo.zhongwen|guo.zhanshe|guo.zhidong|guo.zhaozeng|guo.zhang|guo.zhikun|guo.zuji|guo.zhaomiao|guo.zhiyang|guo.zhenchun|guo.zhisheng|guo.zhitang|guo.zhizheng|guo.zhizhong|guo.zhaobo|guo.zhongning|guo.ziyang|guo.zifang|guo.zhigao|guo.zhongsan|guo.zhan|guo.zhenyan|guo.zhiyuan|guo.zicheng|guo.zhenyu-v|guo.zhifeng|guo.zhurei|guo.zhenghua|guo.zhili|guo.zaiyi|guo.zhiying|guo.zhenwei|guo.zhihao|guo.zhitao|guo.zhenfei|guo.zhenmin|guo.zhongjin|guo.zhenkun|guo.zhijun|guo.zhiwei|guo.zhenfeng|guo.zhilian|guo.zhumei|guo.zhimao|guo.zhao|guo.zhifang|guo.zhaolu|guo.zhiming|guo.zhongkai|guo.zhe|guo.zhaoli|guo.zhiling|guo.zhenyuan|guo.zhian|guo.zusheng|guo.zengyuan|guo.ziyuan|guo.zhanbing|guo.zengxiao|guo.zhenzhen|guo.zhiliao|guo.zichen|guo.zeqing|guo.zhenlin|guo.zhenyong|guo.zhansheng|guo.zili|guo.zhong|guo.zeyu|guo.zoey|guo.zhiyun|guo.zhexiao|guo.zhifen|guo.zhenren|guo.zixue|guo.zhichang|guo.zaoyang|guo.zhanwei|guo.zhengyang|guo.zhiyong|guo.zongqing|guo.zhihong|guo.zichao|guo.zhiqun|guo.zongming|guo.zijun|guo.zede|guo.ziliang|guo.zhanqing|guo.zhimei|guo.zhibo|guo.zhengguang|guo.zhengfei|guo.zhouxiong|guo.zirong|guo.zhixin Consider a tiling of $$\mathbb{R}^2$$ by regular hexagons of unit edge. Let $$H$$ be the set of corners of this tiling. A point of $$H$$ is called an $$H$$-point. An $$H$$-polygon is a simple polygon with vertices in $$H$$. For a polygon $$P$$ denote by $$i_H(P)$$ and $$v_H(P)$$ the number of interior $$H$$-points of $$P$$ and the number of $$H$$-vertices of $$P$$ respectively. Consider the function $G(v) = \min_P\{ i_H(P) \mid v_H(P) =v\},$ where the minimum is taken over all convex $$H$$-polygons. \textit{X.~Feng} et al. [Ars Comb. 120, 321--331 (2015; Zbl 1349.52015)] proved that $$G(v)=0$$ for $$3\leq v\leq 6$$, $$G(7)=G(8)=2$$, $$G(9)$$=4, $$G(10)=6$$. Then \textit{X. Wei} et al. [Ars Comb. 143, 193--203 (2019; Zbl 1463.52015)] obtained that $$10\leq G(11)\leq 12$$. In the present paper the authors show that $$G(12)=12$$. Extending Erdős-Beck's theorem to higher dimensions https://zbmath.org/1472.52022 2021-11-25T18:46:10.358925Z "Do, Thao" https://zbmath.org/authors/?q=ai:do.thao-t Summary: Erdős-Beck's theorem states that $$n$$ points in the plane with at most $$n - x$$ points collinear define at least \textit{cxn} lines for some positive constant $$c$$. It implies $$n$$ points in the plane define $$\Theta ( n^2)$$ lines unless most of the points (i.e. $$n - o(n)$$ points) are collinear. In this paper, we will present two ways to extend this result to higher dimensions. Given a set $$S$$ of $$n$$ points in $$\mathbb{R}^d$$, we want to estimate a lower bound of the number of hyperplanes they define (a hyperplane is defined or spanned by $$S$$ if it contains $$d + 1$$ points of $$S$$ in general position). Our first result says the number of spanned hyperplanes is at least $$c x n^{d - 1}$$ if there exists some hyperplane that contains $$n - x$$ points of $$S$$ and is saturated (as defined in Definition 1.3). Our second result says $$n$$ points in $$\mathbb{R}^d$$ define $$\Theta ( n^d)$$ hyperplanes unless most of the points belong to the union of a collection of flats whose dimension sum to less than $$d$$. Our result has an application to point-hyperplane incidences and a potential application to the point covering problem. Unit distance graphs and algebraic integers https://zbmath.org/1472.52023 2021-11-25T18:46:10.358925Z "Radchenko, Danylo" https://zbmath.org/authors/?q=ai:radchenko.danylo-v Peter Brass posed the following problem: is there a finitely generated additive subgroup of real plane vectors, such that infinitely many elements of the group lies on the unit circle? The problem was motivated by the unit distance problem of Paul Erdős. The paper under review provides a construction for such a subgroup with 4 generators. On the density of the thinnest covering of $$\mathbb{R}^n$$ https://zbmath.org/1472.52024 2021-11-25T18:46:10.358925Z "Jung, Soon-Mo" https://zbmath.org/authors/?q=ai:jung.soon-mo "Nam, Doyun" https://zbmath.org/authors/?q=ai:nam.doyun Let $$K$$ be a convex body in $$\mathbb{R}^n$$. Let $$\mathcal{U}(K)$$ denote the family of all coverings $$\{K + x_i\}_{i\geq 1}$$ of the entire space $$\mathbb{R}^n$$, where $$x_i\in \mathbb{R}^n$$, $$i\geq 1$$. Given $$s>0$$, let $$Q_s = \prod _{i\leq n} [-s, s)$$ and $$R_s = \prod _{i\leq n} (-s, s)$$. The covering density of $$K$$ is defined as $\theta (K) = \inf_{\{K + x_i\}_{i}\in \mathcal{U}(K)} \liminf _{s\to \infty} \frac{1}{\mbox{vol} (Q_s)} \sum_{K+x_i \subset Q_s } \mbox{vol}(K+x_i).$ It is known that $\theta (K) = \inf_{\{K + x_i\}_{i}\in \mathcal{U}(K)} \liminf _{s\to \infty} \frac{1}{\mbox{vol} (Q_s)} \sum_{(K+x_i)\cap Q_s = \emptyset } \mbox{vol} (K+x_i).$ It follows from results of \textit{H.~Groemer} [Math. Z. 81, 260--278 (1963; Zbl 0123.39104)] that for any fixed $$t>0$$, \begin{align*} \theta (K) &= \lim_{s\to \infty }\inf_{\{K + x_i\}_{i}\in \mathcal{U}(K)} \frac{1}{\mbox{vol} (Q_s)} \sum_{(K+x_i)\cap Q_s = \emptyset } \mbox{vol} (K+x_i) \\ &= \lim_{\lambda \to 0^+ }\inf_{\{\lambda K + x_i\}_{i}\in \mathcal{U}(\lambda K)} \frac{1}{\mbox{vol} (R_t)} \sum_{(\lambda K+x_i)\cap R_t = \emptyset } \mbox{vol}(\lambda K+x_i) . \end{align*} The authors provide another proof of this result. They also mention that yet another proof given in [\textit{S.-M. Jung}, Commun. Korean Math. Soc. 10, No. 3, 621--632 (1995, Zbl 0943.52002)] is based on an unverified assertion and contains some ambiguities. Triangular spherical dihedral f-tilings: the $$(\pi/2, \pi/3, \pi/4)$$ and $$(2\pi/3, \pi/4, \pi/4)$$ family https://zbmath.org/1472.52025 2021-11-25T18:46:10.358925Z "Avelino, Catarina P." https://zbmath.org/authors/?q=ai:avelino.catarina-pina "Santos, Altino F." https://zbmath.org/authors/?q=ai:santos.altino-f Edge-to-edge tilings of the Euclidean sphere with the following two properties are studied. They consist of triangles with inner angles $$(\pi/2,\pi/3,\pi/4)$$ or $$(2\pi/3,\pi/4,\pi/4)$$ such that both types appear. They are folding tilings; that is, at each vertex of a tiling there meet an even number of triangles and the sum of alternating angles around it is $$\pi$$. The authors show that there are exactly eight tilings of that kind up to congruence, and they describe their combinatorics and symmetry. Tilings with congruent edge coronae https://zbmath.org/1472.52026 2021-11-25T18:46:10.358925Z "Tomenes, Mark D." https://zbmath.org/authors/?q=ai:tomenes.mark-d "De Las Peñas, Ma. Louise Antonette N." https://zbmath.org/authors/?q=ai:de-las-penas.ma-louise-antonette-n A normal tiling of the Euclidean plane is a cover of the plane by non-overlapping closed topological discs such that their diameters are bounded from above and their inradii are bounded from below by positive constants, and such that the intersection of any two tiles is either empty or a singleton or an arc. Such arcs are called edges of the tiling. A centred edge corona is composed of the centre of an edge and of all tiles having a non-empty intersection with that edge. A tiling is called edge-transitive or isotoxal if its symmetry group acts transitively on the set of all its edges. The authors show that every normal tiling with pairwise congruent centred edge coronae is isotoxal, and they classify such tilings. For the entire collection see [Zbl 1467.52001]. The rhombic triacontahedron and crystallography https://zbmath.org/1472.52027 2021-11-25T18:46:10.358925Z "Senechal, Marjorie" https://zbmath.org/authors/?q=ai:senechal.marjorie-wikler "Taylor, Jean E." https://zbmath.org/authors/?q=ai:taylor.jean-ellen The work is motivated by quasicrystals, whose atomic arrangements ehibit aperiodic order. This overturns the assumption in crystallography that atoms group into tiles and tilings are periodic in all directions. Some cluster models were proposed, which better describe quasicrystals. The paper focuses on the rhombic triacontahedron. The authors propose a more general definition of parallelohedron relaxing the convexity condition. It is shown that some subsets of the rhombic triacontahedron admit tilings of the three-dimensional Euclidean space by translations and correspond to the combinatorial types of lattice Voronoi cells. Also the known types of clusters in icosahedral quasicrystals are unified with the help of some operations applied to the rhombic triacontahedron. For the entire collection see [Zbl 1467.52001]. Control of connectivity and rigidity in prismatic assemblies https://zbmath.org/1472.52028 2021-11-25T18:46:10.358925Z "Choi, Gary P. T." https://zbmath.org/authors/?q=ai:choi.gary-pui-tung "Chen, Siheng" https://zbmath.org/authors/?q=ai:chen.siheng "Mahadevan, L." https://zbmath.org/authors/?q=ai:mahadevan.lakshminarayanan Summary: How can we manipulate the topological connectivity of a three-dimensional prismatic assembly to control the number of internal degrees of freedom and the number of connected components in it? To answer this question in a deterministic setting, we use ideas from elementary number theory to provide a hierarchical deterministic protocol for the control of rigidity and connectivity. We then show that it is possible to also use a stochastic protocol to achieve the same results via a percolation transition. Together, these approaches provide scale-independent algorithms for the cutting or gluing of three-dimensional prismatic assemblies to control their overall connectivity and rigidity. Equivalence of continuous, local and infinitesimal rigidity in normed spaces https://zbmath.org/1472.52029 2021-11-25T18:46:10.358925Z "Dewar, Sean" https://zbmath.org/authors/?q=ai:dewar.sean The author presents a rigorous study of framework rigidity in general finite dimensional normed spaces from the perspective of Lie group actions on smooth manifolds. The results obtained are used to extend the well-known result by L. Asimow and B. Roth establishing the equivalence of local, continuous and infinitesimal rigidity for regular bar-and-joint frameworks in a $$d$$-dimensional Euclidean space, see [Trans. Am. Math. Soc. 245, 279--289 (1978; Zbl 0392.05026)] and [J. Math. Anal. Appl. 68, 171--190 (1979; Zbl 0441.05046)], as well as to obtain upper bounds for the dimension of the space of trivial motions for a framework and establish the flexibility of small frameworks in general non-Euclidean normed spaces. On rigid origami. I: Piecewise-planar paper with straight-line creases https://zbmath.org/1472.52030 2021-11-25T18:46:10.358925Z "He, Zeyuan" https://zbmath.org/authors/?q=ai:he.zeyuan "Guest, Simon D." https://zbmath.org/authors/?q=ai:guest.simon-d Summary: Origami (paper folding) is an effective tool for transforming two-dimensional materials into three-dimensional structures, and has been widely applied to robots, deployable structures, metamaterials, etc. Rigid origami is an important branch of origami where the facets are rigid, focusing on the kinematics of a panel-hinge model. Here, we develop a theoretical framework for rigid origami, and show how this framework can be used to connect rigid origami and its cognate areas, such as the rigidity theory, graph theory, linkage folding and computer science. First, we give definitions regarding fundamental aspects of rigid origami, then focus on how to describe the configuration space of a creased paper. The shape and 0-connectedness of the configuration space are analysed using algebraic, geometric and numeric methods. In the algebraic part, we study the tangent space and generic rigid-foldability based on the polynomial nature of constraints for a panel-hinge system. In the geometric part, we analyse corresponding spherical linkage folding and discuss the special case when there is no cycle in the interior of a crease pattern. In the numeric part, we review methods to trace folding motion and avoid self-intersection. Our results will be instructive for the mathematical and engineering design of origami structures. On rigid origami. II: Quadrilateral creased papers https://zbmath.org/1472.52031 2021-11-25T18:46:10.358925Z "He, Zeyuan" https://zbmath.org/authors/?q=ai:he.zeyuan "Guest, Simon D." https://zbmath.org/authors/?q=ai:guest.simon-d Summary: Miura-ori is well known for its capability of flatly folding a sheet of paper through a tessellated crease pattern made of repeating parallelograms. Many potential applications have been based on the Miura-ori and its primary variations. Here, we are considering how to generalize the Miura-ori: what is the collection of rigid-foldable creased papers with a similar quadrilateral crease pattern as the Miura-ori? This paper reports some progress. We find some new variations of Miura-ori with less symmetry than the known rigid-foldable quadrilateral meshes. They are not necessarily developable or flat-foldable, and still only have single degree of freedom in their rigid folding motion. This article presents a classification of the new variations we discovered and explains the methods in detail. On a fabric of kissing circles https://zbmath.org/1472.52032 2021-11-25T18:46:10.358925Z "Čerňanová, Viera" https://zbmath.org/authors/?q=ai:cernanova.viera Summary: Applying circle inversion on a square grid filled with circles, we obtain a configuration that we call a fabric of kissing circles. We focus on the curvature inside the individual components of the fabric, which are two orthogonal frames and two orthogonal families of chains. We show that the curvatures of the frame circles form a doubly infinite arithmetic sequence (bi-sequence), whereas the curvatures in each chain are arranged in a quadratic bi-sequence. We also prove a sufficient condition for the fabric to be integral. The isostatic conjecture https://zbmath.org/1472.52033 2021-11-25T18:46:10.358925Z "Connelly, Robert" https://zbmath.org/authors/?q=ai:connelly.robert "Gortler, Steven J." https://zbmath.org/authors/?q=ai:gortler.steven-j "Solomonides, Evan" https://zbmath.org/authors/?q=ai:solomonides.evan "Yampolskaya, Maria" https://zbmath.org/authors/?q=ai:yampolskaya.maria Authors' abstract: We show that a jammed packing of disks with generic radii, in a generic container, is such that the minimal number of contacts occurs and there is only one dimension of equilibrium stresses, which have been observed with numerical Monte Carlo simulations. We also point out some connections to packings with different radii and results in the theory of circle packings whose graph forms a triangulation of a given topological surface. Rigidity for sticky discs https://zbmath.org/1472.52034 2021-11-25T18:46:10.358925Z "Connelly, Robert" https://zbmath.org/authors/?q=ai:connelly.robert "Gortler, Steven J." https://zbmath.org/authors/?q=ai:gortler.steven-j "Theran, Louis" https://zbmath.org/authors/?q=ai:theran.louis Summary: We study the combinatorial and rigidity properties of disc packings with generic radii. We show that a packing of $$n$$ discs in the plane with generic radii cannot have more than $$2n - 3$$ pairs of discs in contact. The allowed motions of a packing preserve the disjointness of the disc interiors and tangency between pairs already in contact (modelling a collection of sticky discs). We show that if a packing has generic radii, then the allowed motions are all rigid body motions if and only if the packing has exactly $$2n - 3$$ contacts. Our approach is to study the space of packings with a fixed contact graph. The main technical step is to show that this space is a smooth manifold, which is done via a connection to the Cauchy-Alexandrov stress lemma. Our methods also apply to jamming problems, in which contacts are allowed to break during a motion. We give a simple proof of a finite variant of a recent result of \textit{R. Connelly} et al. [Discrete Comput. Geom. 64, No. 3, 734--758 (2020; Zbl 1472.52033)] on the number of contacts in a jammed packing of discs with generic radii. Exact hyperplane covers for subsets of the hypercube https://zbmath.org/1472.52035 2021-11-25T18:46:10.358925Z "Aaronson, James" https://zbmath.org/authors/?q=ai:aaronson.james "Groenland, Carla" https://zbmath.org/authors/?q=ai:groenland.carla "Grzesik, Andrzej" https://zbmath.org/authors/?q=ai:grzesik.andrzej "Johnston, Tom" https://zbmath.org/authors/?q=ai:johnston.tom "Kielak, Bartłomiej" https://zbmath.org/authors/?q=ai:kielak.bartlomiej The exact cover of $$B\subseteq \{0, 1\}^n$$, denoted by $$\mathrm{ec}(B)$$, is a set of hyperplanes in $$\mathbb{R}^n$$ whose union intersects $$\{0, 1\}^n$$ exactly at $$B$$ -- that is, the points in $$\{0,1\}^n\setminus B$$ are not covered. If just one point, say $$0$$, is removed, then $$\mathrm{ec}(\{0,1\}^n\setminus\{0\})=n$$ from work of \textit{N. Alon} and \textit{Z. Füredi} [Eur. J. Comb. 14, No. 2, 79--83 (1993; Zbl 0773.52011)]. In this article, the authors generalize this to up to $$4$$ points removed. They show that $$\mathrm{ec}(\{0,1\}^n\setminus S) = n-1$$ if $$|S|\in\{2,3\}$$, and if $$|S|=4$$, then $$\mathrm{ec}(\{0,1\}^n\setminus S)$$ is $$n-1$$ if the four points of $$S$$ are not coplanar, and $$n-2$$ otherwise. The authors prove asymptotic bounds concerning the following two numbers: for $$n,k\in\mathbb{N}$$, \begin{align*} \mathrm{ec}(n, k) &= \max\{\mathrm{ec}(\{0,1\}^n\setminus S) : S\subseteq \{0,1\}^n, \; |S|=k \}, \\ \mathrm{ec}(n) &= \max\{\mathrm{ec}(B) : B\subseteq \{0,1\}^n \}. \end{align*} They close by posing two problems that would, if answered affirmatively, tighten the bounds they determined. On the cut number problem for the 4, and 5-cubes https://zbmath.org/1472.52036 2021-11-25T18:46:10.358925Z "Emamy-K, M. R." https://zbmath.org/authors/?q=ai:emamy-k.m-reza "Arce-Nazario, R." https://zbmath.org/authors/?q=ai:arce-nazario.rafael-a Summary: The hypercube cut number $$S ( d )$$ is the minimum number of hyperplanes in the $$d$$-dimensional Euclidean space that slice all the edges of the $$d$$-cube. The determination of $$S ( d )$$ in dimensions 5, 6, and 7, is one of the Victor Klee's unresolved problems presented in one of his invited talks on problems from discrete geometry. The value of $$S ( d )$$ is unknown for $$d \geq 7$$, but for $$d \leq 3$$ it is trivial to show $$S ( d ) = d$$. In the late nineteen eighties, Emamy-K has shown $$S ( 4 ) = 4$$ via two different proofs. More than a decade later, Sohler and Ziegler obtained a computational proof of $$S ( 5 ) = 5$$ that took about two months of CPU computing time. From $$S ( 5 ) = 5$$ it can be verified that $$S ( 6 ) = 5$$. A short mathematical proof for $$d = 5$$ remains to be a challenging problem that also leads one to the insight of the still open 7-dimensional problem. In this article we present a vertex coloring approach on the hypercube that gives a simplified proof for $$d = 4$$ and also helps toward a short mathematical proof for $$d = 5$$, free of computer computations. Worpitzky-compatible subarrangements of braid arrangements and cocomparability graphs https://zbmath.org/1472.52037 2021-11-25T18:46:10.358925Z "Tran, Tan Nhat" https://zbmath.org/authors/?q=ai:tran.tan-nhat "Tsuchiya, Akiyoshi" https://zbmath.org/authors/?q=ai:tsuchiya.akiyoshi Summary: The class of Worpitzky-compatible subarrangements of a Weyl arrangement together with an associated Eulerian polynomial was recently introduced by \textit{A. U. Ashraf} et al. [Adv. Appl. Math. 120, Article ID 102064, 24 p. (2020; Zbl 1447.52026)], which brings the characteristic and Ehrhart quasi-polynomials into one formula. The subarrangements of the braid arrangement, the Weyl arrangement of type $$A$$, are known as the graphic arrangements. We prove that the Worpitzky-compatible graphic arrangements are characterized by cocomparability graphs. This can be regarded as a counterpart of the characterization by Stanley and Edelman-Reiner of free and supersolvable graphic arrangements in terms of chordal graphs. Our main result yields new formulas for the chromatic and graphic Eulerian polynomials of cocomparability graphs. Quadrilaterals inscribed in convex curves https://zbmath.org/1472.53007 2021-11-25T18:46:10.358925Z "Matschke, Benjamin" https://zbmath.org/authors/?q=ai:matschke.benjamin Summary: We classify the set of quadrilaterals that can be inscribed in convex Jordan curves, in the continuous as well as in the smooth case. This answers a question of Makeev in the special case of convex curves. The difficulty of this problem comes from the fact that standard topological arguments to prove the existence of solutions do not apply here due to the lack of sufficient symmetry. Instead, the proof makes use of an area argument of Karasev and Tao, which we furthermore simplify and elaborate on. The continuous case requires an additional analysis of the singular points, and a small miracle, which then extends to show that the problems of inscribing isosceles trapezoids in smooth curves and in piecewise $$C^1$$ curves are equivalent. Nonsmooth convexity and monotonicity in terms of a bifunction on Riemannian manifolds https://zbmath.org/1472.53049 2021-11-25T18:46:10.358925Z "Ansari, Qamrul Hasan" https://zbmath.org/authors/?q=ai:ansari.qamrul-hasan "Islam, Monirul" https://zbmath.org/authors/?q=ai:islam.monirul "Yao, Jen-Chih" https://zbmath.org/authors/?q=ai:yao.jen-chih Summary: In this paper, we introduce geodesic $$h$$-convexity, geodesic $$h$$-pseudoconvexity and geodesic $$h$$-quasiconvexity of a real-valued function defined on a geodesic convex subset of a Riemannian manifold in terms of a bifunction $$h$$. We extend Diewart's mean value theorem for Dini directional derivatives to the Riemannian manifolds. By using this mean value theorem, we present some relations between geodesic convexity and geodesic $$h$$-convexity, geodesic pseudoconvexity and geodesic $$h$$-pseudoconvexity, and geodesic quasiconvexity and geodesic $$h$$-quasiconvexity. We also introduce monotonicity, quasimonotonicity and pseudomonotonicity for the bifunction $$h$$. We investigate the relations between geodesic $$h$$-convexity of a real-valued function and monotonicity of $$h$$, geodesic $$h$$-pseudoconvexity of a real-valued function and pseudomonotonicity of $$h$$, and geodesic $$h$$-quasiconvexity of a real-valued function and quasimonotonicity of $$h$$. We introduce the geodesic $$h$$-pseudolinearity of a real-valued function defined on geodesic convex subset of a Riemannian manifold. We provide some characterizations of geodesic $$h$$-pseudolinearity, and give some relations between geodesic $$h$$-pseudolinearity and geodesic pseudolinearity. The pseudoaffiness of a bifunction $$h$$ is introduced and some of its characterizations are also presented. The dual Bonahon-Schläfli formula https://zbmath.org/1472.53086 2021-11-25T18:46:10.358925Z "Mazzoli, Filippo" https://zbmath.org/authors/?q=ai:mazzoli.filippo In the paper, for a differentiable deformation of geometrically finite 3-manifold the Bonahon-Schläfli Formula gives the derivative of the volume of the convex cones in terms of the variation of the geometry of their boundaries (the classical Schläfli Formula is applicable for determining the volume of the hyperbolic polyhedra). Here the author studies the analogous problem for the dual volume and gives a self-contained proof of the dual Bonahon-Schläfli Formula (without using Bonahon's results). Combinatorial $$p$$-th Ricci flows on surfaces https://zbmath.org/1472.53103 2021-11-25T18:46:10.358925Z "Lin, Aijin" https://zbmath.org/authors/?q=ai:lin.aijin "Zhang, Xiaoxiao" https://zbmath.org/authors/?q=ai:zhang.xiaoxiao Summary: For any $$p>1$$ and triangulated surfaces, we introduce the combinatorial $$p$$-th Ricci flow which exactly equals the combinatorial Ricci flow first introduced by \textit{B. Chow} and \textit{F. Luo} [J. Differ. Geom. 63, No. 1, 97--129 (2003; Zbl 1070.53040)] when $$p=2$$. Then we show the long time existence and convergence of the solution to the combinatorial $$p$$-th Ricci flow. Our results partially generalize Chow-Luo's work on the combinatorial Ricci flow from $$p=2$$ to any $$p>1$$. Contraction principle for trajectories of random walks and Cramér's theorem for kernel-weighted sums https://zbmath.org/1472.60053 2021-11-25T18:46:10.358925Z "Vysotsky, Vladislav" https://zbmath.org/authors/?q=ai:vysotsky.vladislav-v Summary: In 2013 \textit{A. A. Borovkov} and \textit{A. A. Mogulskii} [Theory Probab. Appl. 57, No. 1, 1--27 (2013; Zbl 1279.60037); translation from Teor. Veroyatn. Primen. 57, No. 1, 3--34 (2012)] proved a weaker-than-standard metric'' large deviations principle (LDP) for trajectories of random walks in $$\mathbb{R}^d$$ whose increments have the Laplace transform finite in a neighbourhood of zero. We prove that general metric LDPs are preserved under uniformly continuous mappings. This allows us to transform the result of Borovkov and Mogulskii into standard LDPs. We also give an explicit integral representation of the rate function they found. As an application, we extend the classical Cramér theorem by proving an LPD for kernel-weighted sums of i.i.d. random vectors in $$\mathbb{R}^d$$. Maximin distance designs based on densest packings https://zbmath.org/1472.62125 2021-11-25T18:46:10.358925Z "Yang, Liuqing" https://zbmath.org/authors/?q=ai:yang.liuqing "Zhou, Yongdao" https://zbmath.org/authors/?q=ai:zhou.yongdao "Liu, Min-Qian" https://zbmath.org/authors/?q=ai:liu.min-qian Summary: Computer experiments play a crucial role when physical experiments are expensive or difficult to be carried out. As a kind of designs for computer experiments, maximin distance designs have been widely studied. Many existing methods for obtaining maximin distance designs are based on stochastic algorithms, and these methods will be infeasible when the run size or number of factors is large. In this paper, we propose some deterministic construction methods for maximin $$L_2$$-distance designs in two to five dimensions based on densest packings. The resulting designs have large $$L_2$$-distances and are mirror-symmetric. Some of them have the same $$L_2$$-distances as the existing optimal maximin distance designs, and some of the others are completely new. Especially, the resulting 2-dimensional designs possess a good projection property. An optimal algorithm for tiling the plane with a translated polyomino https://zbmath.org/1472.68208 2021-11-25T18:46:10.358925Z "Winslow, Andrew" https://zbmath.org/authors/?q=ai:winslow.andrew Summary: We give a $$O(n)$$-time algorithm for determining whether translations of a polyomino with $$n$$ edges can tile the plane. The algorithm is also a $$O(n)$$-time algorithm for enumerating all regular tilings, and we prove that at most $$\varTheta (n)$$ such tilings exist. For the entire collection see [Zbl 1326.68015]. Big influence of small random imperfections in origami-based metamaterials https://zbmath.org/1472.74013 2021-11-25T18:46:10.358925Z "Liu, Ke" https://zbmath.org/authors/?q=ai:liu.ke "Novelino, Larissa S." https://zbmath.org/authors/?q=ai:novelino.larissa-s "Gardoni, Paolo" https://zbmath.org/authors/?q=ai:gardoni.paolo "Paulino, Glaucio H." https://zbmath.org/authors/?q=ai:paulino.glaucio-h Summary: Origami structures demonstrate great theoretical potential for creating metamaterials with exotic properties. However, there is a lack of understanding of how imperfections influence the mechanical behaviour of origami-based metamaterials, which, in practice, are inevitable. For conventional materials, imperfection plays a profound role in shaping their behaviour. Thus, this paper investigates the influence of small random geometric imperfections on the nonlinear compressive response of the representative Miura-ori, which serves as the basic pattern for many metamaterial designs. Experiments and numerical simulations are used to demonstrate quantitatively how geometric imperfections hinder the foldability of the Miura-ori, but on the other hand, increase its compressive stiffness. This leads to the discovery that the residual of an origami foldability constraint, given by the Kawasaki theorem, correlates with the increase of stiffness of imperfect origami-based metamaterials. This observation might be generalizable to other flat-foldable patterns, in which we address deviations from the zero residual of the perfect pattern; and to non-flat-foldable patterns, in which we would address deviations from a finite residual. Inflationary routes to Gaussian curved topography https://zbmath.org/1472.74154 2021-11-25T18:46:10.358925Z "Siéfert, Emmanuel" https://zbmath.org/authors/?q=ai:siefert.emmanuel "Warner, Mark" https://zbmath.org/authors/?q=ai:warner.mark-r-e Summary: Gaussian-curved shapes are obtained by inflating initially flat systems made of two superimposed strong and light thermoplastic impregnated fabric sheets heat-sealed together along a specific network of lines. The resulting inflated structures are light and very strong because they (largely) resist deformation by the intercession of stretch. Programmed patterns of channels vary either discretely through boundaries or continuously. The former give rise to faceted structures that are in effect non-isometric origami and that cannot unfold as in conventional folded structures since they present the localized angle deficit or surplus. Continuous variation of the channel direction in the form of spirals is examined, giving rise to curved shells. We solve the inverse problem consisting in finding a network of seam lines leading to a target axisymmetric shape on inflation. They too have strength from the metric changes that have been pneumatically driven, resistance to change being met with stretch and hence high forces like typical shells. Evolution of local motifs and topological proximity in self-assembled quasi-crystalline phases https://zbmath.org/1472.82045 2021-11-25T18:46:10.358925Z "Pedersen, Martin Cramer" https://zbmath.org/authors/?q=ai:pedersen.martin-cramer "Robins, Vanessa" https://zbmath.org/authors/?q=ai:robins.vanessa "Mortensen, Kell" https://zbmath.org/authors/?q=ai:mortensen.kell "Kirkensgaard, Jacob J. K." https://zbmath.org/authors/?q=ai:kirkensgaard.jacob-j-k Summary: Using methods from the field of topological data analysis, we investigate the self-assembly and emergence of three-dimensional quasi-crystalline structures in a single-component colloidal system. Combining molecular dynamics and persistent homology, we analyse the time evolution of persistence diagrams and particular local structural motifs. Our analysis reveals the formation and dissipation of specific particle constellations in these trajectories, and shows that the persistence diagrams are sensitive to nucleation and convergence to a final structure. Identification of local motifs allows quantification of the similarities between the final structures in a topological sense. This analysis reveals a continuous variation with density between crystalline clathrate, quasi-crystalline, and disordered phases quantified by topological proximity', a visualization of the Wasserstein distances between persistence diagrams. From a topological perspective, there is a subtle, but direct connection between quasi-crystalline, crystalline and disordered states. Our results demonstrate that topological data analysis provides detailed insights into molecular self-assembly. Robust Farkas-Minkowski constraint qualification for convex inequality system under data uncertainty https://zbmath.org/1472.90078 2021-11-25T18:46:10.358925Z "Li, Xiao-Bing" https://zbmath.org/authors/?q=ai:li.xiaobing "Al-Homidan, Suliman" https://zbmath.org/authors/?q=ai:al-homidan.suliman-s "Ansari, Qamrul Hasan" https://zbmath.org/authors/?q=ai:ansari.qamrul-hasan "Yao, Jen-Chih" https://zbmath.org/authors/?q=ai:yao.jen-chih The authors consider the nonempty assumed solution set $$S=\{x\in \mathbb{R}^n \mid g_i(x,u_i)\le 0\; \forall\,u_i\in \mathcal{U}_i,\, i\in \mathcal{I}=1,2,\dots k \}$$ of a robust finite inequality system with compact convex uncertainty sets $$\mathcal{U}_i$$ and convex-concave finite valued functions $$g_i:\mathbb{R}^n\times\mathbb{R}^m\rightarrow\mathbb{R}$$, $$i\in \mathcal{I}$$. They show (Th. 1) that the robust global error bound (RGEB) $$\alpha \inf_{y\in S}\|x-y\|\le \sum_{i\in \mathcal{I}}[\sup_{u_i\in \mathcal{U}_i}g_i(x,u_i)]_+$$ on $$\mathbb{R}^n$$ for some $$\alpha>0$$ is sufficient for the validity of the robust Farkas-Minkowski constraint qualification (FMCQ) according to $$\mathrm{epi}\,\delta^*_S=\bigcup_{\lambda_i>0,\,\,u_i\in \mathcal{U}_i,\,i\in \mathcal{I}}\mathrm{epi}\,(\sum_{i\in \mathcal{I}}\lambda_ig_i(x,u_i))^*$$. $$\mathrm{epi} f$$ is the epigraph of an extended real-valued function $$f$$ and $$\delta^*_S$$ denotes the support functional of the convex set $$S$$. Hence the union is a closed convex cone. Example 3.2 demonstrates that the concavity of $$g_i$$ w.r.t. $$u_i$$ is essential. Conditions (FMCQ) and (REGB) are equivalent for the special case of in $$x$$ positively semidefinite quadratic forms $$g_i$$ with coefficients $$u_i$$ belonging to a scenario uncertainty set. In the last three rows of the proof of Th. 1, the subset sign should be replaced by the corresponding superset sign for getting equality. A new penalty/stochastic approach to an application of the covering problem: the gamma knife treatment https://zbmath.org/1472.90116 2021-11-25T18:46:10.358925Z "Venceslau, Marilis Bahr Karam" https://zbmath.org/authors/?q=ai:venceslau.marilis-bahr-karam "Venceslau, Helder Manoel" https://zbmath.org/authors/?q=ai:venceslau.helder-manoel "Pinto, Renan Vicente" https://zbmath.org/authors/?q=ai:pinto.renan-vicente "Dias, Gustavo" https://zbmath.org/authors/?q=ai:dias.gustavo-fruet "Maculan, Nelson" https://zbmath.org/authors/?q=ai:maculan.nelson-f Summary: The covering problem of a three dimensional body using different radii spheres is considered. The motivating application -- the treatment planning of Gamma Knife radiosurgery -- is briefly discussed. We approach the problem only by the geometric covering point of view, that is, given a set of spheres and a body, the objective is to cover the body using the smallest possible number of spheres, regardless of the dosage issue. In order to solve this mathematical programming problem, we consider an approach based on the application of penalty and stochastic local search techniques. Finally, some illustrative results and comparisons are presented. An iterative vertex enumeration method for objective space based vector optimization algorithms https://zbmath.org/1472.90121 2021-11-25T18:46:10.358925Z "Kaya, İrfan Caner" https://zbmath.org/authors/?q=ai:kaya.irfan-caner "Ulus, Firdevs" https://zbmath.org/authors/?q=ai:ulus.firdevs Summary: An application area of vertex enumeration problem (VEP) is the usage within objective space based linear/convex vector optimization algorithms whose aim is to generate (an approximation of) the Pareto frontier. In such algorithms, VEP, which is defined in the objective space, is solved in each iteration and it has a special structure. Namely, the recession cone of the polyhedron to be generated is the ordering cone. We consider and give a detailed description of a vertex enumeration procedure, which iterates by calling a modified `double description (DD) method'' that works for such unbounded polyhedrons. We employ this procedure as a function of an existing objective space based vector optimization algorithm (Algorithm 1); and test the performance of it for randomly generated linear multiobjective optimization problems. We compare the efficiency of this procedure with another existing DD method as well as with the current vertex enumeration subroutine of Algorithm 1. We observe that the modified procedure excels the others especially as the dimension of the vertex enumeration problem (the number of objectives of the corresponding multiobjective problem) increases.