Points of an orbit of a finite Coxeter group G, generated by n reflections starting from a single seed point, are considered as vertices of a polytope (G-polytope) centered at the origin of a real n-dimensional Euclidean space. A general efficient method is recalled for the geometric description of G- polytopes, their faces of all dimensions and their adjacencies. Products and symmetrized powers of G-polytopes are introduced and their decomposition into the sums of G-polytopes is described. Several invariants of G-polytopes are found, namely the analogs of Dynkin indices of degrees 2 and 4, anomaly numbers and congruence classes of the polytopes. The definitions apply to crystallographic and non-crystallographic Coxeter groups. Examples and applications are shown.
We show that there exist reduced polytopes in three-dimensional Euclidean space. This partially answers the question posed by Lassak on the existence of reduced polytopes in $d$-dimensional Euclidean space for $d\geq 3$. Moreover, we prove a novel necessary condition on reduced polytopes in three-dimensional Euclidean space.
Using equivariant topology, we prove that it is always possible to find $n$ points in the $d$-dimensional faces of a $nd$-dimensional convex polytope $P$ so that their center of mass is a target point in $P$. Equivalently, the $n$-fold Minkowski sum of a $nd$-polytope's $d$-skeleton is that polytope scaled by $n$. This verifies a conjecture by Takeshi Tokuyama.
This note is a comment to the paper by D.R.Heath-Brown and B.Z.Moroz (Math Proc. Camb. Phil. Soc. 125 (1999)). That paper concerns with the projective surface $S$ in $\mathbb{P}^{3}$ defined by the equation $x_{1}x_{2}x_{3}=x_{4}^{3}$. It is shown there that the evaluation of the leading term of the asymptotic formula for the number of rational points of bounded height in $S(\Q)$ is equivalent to the evaluation of the volume of some 6-dimensional polytope $¶$. The volume of $¶$ is known from several papers; we calculate this volume by elementary method using symmetry of the polytope $¶$. We al…
A polytope in a finite-dimensional normed space is subequilateral if the length in the norm of each of its edges equals its diameter. Subequilateral polytopes occur in the study of two unrelated subjects: surface energy minimizing cones and edge-antipodal polytopes. We show that the number of vertices of a subequilateral polytope in any d-dimensional normed space is bounded above by (d/2+1)^d for any d >= 2. The same upper bound then follows for the number of vertices of the edge-antipodal polytopes introduced by I.Talata (Period. Math. Hungar. 38 (1999), 231--246). This is a constructive impr…
By correcting an example by Polyanskii, we show that there exist reduced polytopes in three-dimensional Euclidean space. This partially answers the question posed by Lassak on the existence of reduced polytopes in $d$-dimensional Euclidean space for $d\geq 3$.
For a lattice $L$ of $R^n$, a sphere $S(c,r)$ of center $c$ and radius $r$ is called {\em empty} if for any $v\in L$ we have $\Vert v - c\Vert \geq r$. Then the set $S(c,r)\cap L$ is the vertex set of a {\em Delaunay polytope} $P=conv(S(c,r)\cap L)$. A Delaunay polytope is called {\em perfect} if any affine transformation $ϕ$ such that $ϕ(P)$ is a Delaunay polytope is necessarily an isometry of the space composed with an homothety. Perfect Delaunay polytopes are remarkable structure that exist only if $n=1$ or $n\geq 6$ and they have shown up recently in covering maxima studies. Here we give a…
Let $P\subset\R^d$ be a $d$-dimensional polytope. The {\em realization space} of~$P$ is the space of all polytopes $P'\subset\R^d$ that are combinatorially equivalent to~$P$, modulo affine transformations. We report on work by the first author, which shows that realization spaces of \mbox{4-dimensional} polytopes can be ``arbitrarily bad'': namely, for every primary semialgebraic set~$V$ defined over~$\Z$, there is a $4$-polytope $P(V)$ whose realization space is ``stably equivalent'' to~$V$. This implies that the realization space of a $4$-polytope can have the homotopy type of an arbitrary f…
The $f$-vector of a polytope consists of the numbers of its $i$-dimensional faces. An open field of study is the characterization of all possible $f$-vectors. It has been solved in three dimensions by Steinitz in the early 19th century. We state a related question, i.e. to characterize $f$-vectors of three dimensional polytopes respecting a symmetry, given by a finite group of matrices. We give a full answer for all three dimensional polytopes that are symmetric with respect to a finite rotation or rotary reflection group. We solve these cases constructively by developing tools that generalize…
The packing density of the $n$-dimensional cross-polytope
Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences（匈牙利科学院阿尔弗雷德·雷尼数学研究所）；Bolyai Institute, University of Szeged（塞格德大学博莱研究所）；University of Calgary（卡尔加里大学）
In 3-dimensional Euclidean space there exist two exceptional polyhedra, the rhombic dodecahedron and the rhombic triacontahedron, the only known polytopes (besides polygons) that are edge-transitive without being vertex-transitive. We show that these polyhedra do not have higher-dimensional analogues, that is, that in dimension $d\ge 4$, edge-transitivity of convex polytopes implies vertex-transitivity. More generally, we give a classification of all convex polytopes which at the same time have all edges of the same length, an edge in-sphere and a bipartite edge-graph. We show that any such po…
We study properties of the volume of projections of the $n$-dimensional cross-polytope $\crosp^n = \{ x \in \R^n \mid |x_1| + \dots + |x_n| \leqslant 1\}.$ We prove that the projection of $\crosp^n$ onto a $k$-dimensional coordinate subspace has the maximum possible volume for $k=2$ and for $k=3.$ We obtain the exact lower bound on the volume of such a projection onto a two-dimensional plane. Also, we show that there exist local maxima which are not global ones for the volume of a projection of $\crosp^n$ onto a $k$-dimensional subspace for any $n > k \geqslant 2.$
In the spirit of the Genetics of the Regular Figures, by L. Fejes Tóth, we prove the following theorem: If $2n$ points are selected in the $n$-dimensional Euclidean ball $B^n$ so that the smallest distance between any two of them is as large as possible, then the points are the vertices of an inscribed regular cross-polytope. This generalizes a result of R. A. Rankin for $2n$ points on the surface of the ball. We also generalize, in the same manner, a theorem of Davenport and Hajós on a set of $n+2$ points. As a corollary, we obtain a solution to the problem of packing $k$ unit $n$-dimensional…
Based on Minkowski's work on critical lattices of 3-dimensional convex bodies we present an efficient algorithm for computing the density of a densest lattice packing of an arbitrary 3-polytope. As an application we calculate densest lattice packings of all regular and Archimedean polytopes.
For $3$-dimensional convex polytopes, inscribability is a classical property that is relatively well-understood due to its relation with Delaunay subdivisions of the plane and hyperbolic geometry. In particular, inscribability can be tested in polynomial time, and for every $f$-vector of $3$-polytopes, there exists an inscribable polytope with that $f$-vector. For higher-dimensional polytopes, much less is known. Of course, for any inscribable polytope, all of its lower-dimensional faces need to be inscribable, but this condition does not appear to be very strong. We observe non-trivial new ob…
Given a lattice $L$, a full dimensional polytope $P$ is called a {\em Delaunay polytope} if the set of its vertices is $S\cap L$ with $S$ being an {\em empty sphere} of the lattice. Extending our previous work \cite{DD-hyp} on the {\em hypermetric cone} $HYP_7$, we classify the six-dimensional Delaunay polytopes according to their {\em combinatorial type}. The list of 6241 combinatorial types is obtained by a study of the set of faces of the polyhedral cone $HYP_7$.
Let $Q_n$ denote the $n$-dimensional hypercube with the vertex set $V_n=\{0,1}^n$. A 0/1-polytope of $Q_n$ is a convex hull of a subset of $V_n$. This paper is concerned with the enumeration of equivalence classes of full-dimensional 0/1-polytopes under the symmetries of the hypercube. With the aid of a computer program, Aichholzer completed the enumeration of equivalence classes of full-dimensional 0/1-polytopes for $Q_4$, $Q_5$, and those of $Q_6$ up to 12 vertices. In this paper, we present a method to compute the number of equivalence classes of full-dimensional 0/1-polytopes of $Q_n$ with…
A polynomial representation of a convex d-polytope P is a finite set \{p_1(x),...,p_n(x)\} of polynomials over E^d such that P=\setcond{x \in \E^d}{p_1(x) \ge 0 {for every} 1 \le i \le n}. By s(d,P) we denote the least possible number of polynomials in a polynomial representation of P. It is known that d \le s(d,P) \le 2d-1. Moreover, it is conjectured that s(d,P)=d for all convex d-polytopes P. We confirm this conjecture for simple d-polytopes by providing an explicit construction of d polynomials that represent a given simple d-polytope P.
The beta polytope $P_{n,d}^β$ is the convex hull of $n$ i.i.d. random points distributed in the unit ball of $\mathbb{R}^d$ according to a density proportional to $(1-\lVert{x}\rVert^2)^β$ if $β>-1$ (in particular, $β=0$ corresponds to the uniform distribution in the ball), or uniformly on the unit sphere if $β=-1$. We show that the expected normalized volumes of high-dimensional beta polytopes exhibit a phase transition and we describe its shape. We derive analogous results for the intrinsic volumes of beta polytopes and, when $β=0$, their number of vertices.
Let $X_1,\ldots,X_N$, $N>n$, be independent random points in $\mathbb{R}^n$, distributed according to the so-called beta or beta-prime distribution, respectively. We establish threshold phenomena for the volume, intrinsic volumes, or more general measures of the convex hulls of these random point sets, as the space dimension $n$ tends to infinity. The dual setting of polytopes generated by random halfspaces is also investigated.
We present an interactive game which challenges a single player to match 3-dimensional polytopes to their planar nets. It is open source, and it runs in standard web browsers
We show that every heptagon is a section of a $3$-polytope with $6$ vertices. This implies that every $n$-gon with $n\geq 7$ can be obtained as a section of a $(2+\lfloor\frac{n}{7}\rfloor)$-dimensional polytope with at most $\lceil\frac{6n}{7}\rceil$ vertices; and provides a geometric proof of the fact that every nonnegative $n\times m$ matrix of rank $3$ has nonnegative rank not larger than $\lceil\frac{6\min(n,m)}{7}\rceil$. This result has been independently proved, algebraically, by Shitov (J. Combin. Theory Ser. A 122, 2014).
Let $P_n$ be an $n$-dimensional regular polytope from one of the three infinite series (regular simplices, regular crosspolytopes, and cubes). Project $P_n$ onto a random, uniformly distributed linear subspace of dimension $d\geq 2$. We prove that the expected number of $k$-dimensional faces of the resulting random polytope is an increasing function of $n$. As a corollary, we show that the expected number of $k$-faces of the Gaussian polytope is an increasing function of the number of points used to generate the polytope. Similar results are obtained for the symmetric Gaussian polytope and the…
The Monotone Upper Bound Problem (Klee, 1965) asks if the number M(d,n) of vertices in a monotone path along edges of a d-dimensional polytope with n facets can be as large as conceivably possible: Is M(d,n) = M_{ubt}(d,n), the maximal number of vertices that a d-polytope with n facets can have according to the Upper Bound Theorem? We show that in dimension d=4, the answer is ``yes'', despite the fact that it is ``no'' if we restrict ourselves to the dual-to-cyclic polytopes. For each n>=5, we exhibit a realization of a polar-to-neighborly 4-dimensional polytope with n facets and a Hamilton pa…
We show that up to unimodular equivalence there are only finitely many d-dimensional lattice polytopes without interior lattice points that do not admit a lattice projection onto a (d-1)-dimensional lattice polytope without interior lattice points. This was conjectured by Treutlein. As an immediate corollary, we get a short proof of a recent result of Averkov, Wagner and Weismantel, namely the finiteness of the number of maximal lattice polytopes without interior lattice points. Moreover, we show that in dimension four and higher some of these finitely many polytopes are not maximal as convex …
Let $\mathcal{P}$ and $\mathcal{P}'$ be $3$-dimensional convex polytopes in $\mathbb{R}^3$ and $S \subseteq \mathbb{R}^3$ be a non-empty intersection of an open set with a sphere. As a consequence of a somewhat more general result it is proved that $\mathcal{P}$ and $\mathcal{P}'$ coincide up to translation and/or reflection in a point if $|\int_{\mathcal{P}} e^{-i\mathbf{s}\cdot\mathbf{x}} \,\mathbf{dx}| = |\int_{\mathcal{P}'} e^{-i\mathbf{s}\cdot\mathbf{x}} \,\mathbf{dx}|$ for all $\mathbf{s} \in S$. This can be applied to the field of crystallography regarding the question whether a nanopar…
Suppose we choose $N$ points uniformly randomly from a convex body in $d$ dimensions. How large must $N$ be, asymptotically with respect to $d$, so that the convex hull of the points is nearly as large as the convex body itself? It was shown by Dyer-Füredi-McDiarmid that exponentially many samples suffice when the convex body is the hypercube, and by Pivovarov that the Euclidean ball demands roughly $d^{d/2}$ samples. We show that when the convex body is the simplex, exponentially many samples suffice; this then implies the same result for any convex simplicial polytope with at most exponentia…
We describe an algorithm to enumerate polytopes. This algorithm is then implemented to give a complete classification of combinatorial spheres of dimension 3 with 9 vertices and decide polytopality of those spheres. In particular, we completely enumerate all combinatorial types of 4-dimensional polytopes with 9 vertices. It is shown that all of those combinatorial types are rational: They can be realized with rational coordinates. We find 316014 combinatorial spheres on 9 vertices. Of those, 274148 can be realized as the boundary complex of a four-dimensional polytope and the remaining 41866 a…
We give an upper bound on the volume vol(P*) of a polytope P* dual to a d-dimensional lattice polytope P with exactly one interior lattice point, in each dimension d. This bound, expressed in terms of the Sylvester sequence, is sharp, and is achieved by the dual to a particular reflexive simplex. Our result implies a sharp upper bound on the volume of a d-dimensional reflexive polytope. Translated into toric geometry, this gives a sharp upper bound on the anti-canonical degree $(-K_X)^d$ of a d-dimensional toric Fano variety X with at worst canonical singularities.
The cross covariogram g_{K,L} of two convex sets K, L in R^n is the function which associates to each x in R^n the volume of the intersection of K with L+x. The problem of determining the sets from their covariogram is relevant in stochastic geometry, in probability and it is equivalent to a particular case of the phase retrieval problem in Fourier analysis. It is also relevant for the inverse problem of determining the atomic structure of a quasicrystal from its X-ray diffraction image. The two main results of this paper are that g_{K,K} determines three-dimensional convex polytopes K and tha…
It is well known that R^N has subspaces of dimension proportional to N on which the \ell_1 norm is equivalent to the \ell_2 norm; however, no explicit constructions are known. Extending earlier work by Artstein--Avidan and Milman, we prove that such a subspace can be generated using O(N) random bits.
Let $2\le k\le d-1$ and let $P$ and $Q$ be two convex polytopes in ${\mathbb E^d}$. Assume that their projections, $P|H$, $Q|H$, onto every $k$-dimensional subspace $H$, are congruent. In this paper we show that $P$ and $Q$ or $P$ and $-Q$ are translates of each other. We also prove an analogous result for sections by showing that $P=Q$ or $P=-Q$, provided the polytopes contain the origin in their interior and their sections, $P \cap H$, $Q \cap H$, by every $k$-dimensional subspace $H$, are congruent.
