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）
The packing density of the regular cross-polytope in Euclidean $n$-space is unknown except in dimensions $2$ and $4$ where it is 1. The only non-trivial upper bound is due to Gravel, Elser, and Kallus (2011) who proved that for $n=3$ the packing density of the regular octahedron is at most $1-1.4\ldots\times 10^{-12}$. In this paper, we prove upper bounds for the packing density of the $n$-dimensional regular cross-polytope in the case that $n\geq 7$. We use a modification of Blichfeldt's method due to G. Fejes Tóth and W. Kuperberg (1993).
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.
It is well known that the vertices of any Euclidean simplicial regular polytope determine an optimal packing of equal spherical balls. We prove a stability version of optimal order of this result.
The problem of finding the largest number of points in the unit cross-polytope such that the $l_{1}$-distance between any two distinct points is at least $2r$ is investigated for $r\in\left(1-\frac{1}{n},1\right]$ in dimensions $\geq2$ and for $r\in\left(\frac{1}{2},1\right]$ in dimension $3$. For the $n$-dimensional cross-polytope, $2n$ points can be placed when $r\in\left(1-\frac{1}{n},1\right]$. For the three-dimensional cross-polytope, $10$ and $12$ points can be placed if and only if $r\in\left(\frac{3}{5},\frac{2}{3}\right]$ and $r\in\left(\frac{4}{7},\frac{3}{5}\right]$ respectively, an…
We investigate in this paper the relation between Apollonian $d$-ball packings and stacked $(d+1)$-polytopes for dimension $d\ge 3$. For $d=3$, the relation is fully described: we prove that the $1$-skeleton of a stacked $4$-polytope is the tangency graph of an Apollonian $3$-ball packing if and only if no six $4$-cliques share a $3$-clique. For higher dimension, we have some partial results.
Two constructions of lattice packings of $ n $-dimensional cross-polytopes ($ \ell_1 $ balls) are described, the density of which exceeds that of any prior construction by a factor of at least $ 2^{\frac{n}{\ln n}(1 + o(1))} $ when $ n \to \infty $. The first family of lattices is explicit and is obtained by applying Construction A to a class of Reed-Solomon codes. The second family has subexponential construction complexity and is based on the notion of Sidon sets in finite Abelian groups. The construction based on Sidon sets also gives the highest known asymptotic density of packing discrete…
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…
In 1957, Hadwiger made a conjecture that every $n$-dimensional convex body can be covered by $2^n$ translations of its interior. The Hadwiger's covering functional $γ_m(K)$ is the smallest positive number $r$ such that $K$ can be covered by $m$ translations of $rK$. Due to Zong's program, we study the Hadwiger's covering functional for the simplex and the cross-polytope. In this paper, we give upper bounds for the Hadwiger's covering functional of the simplex and the cross-polytope.
We find minimal and maximal length of intersections of lines at a fixed distance to the origin with the cross-polytope. We also find maximal volume noncentral sections of the cross-polypote by hyperplanes which are at a fixed large distance to the origin and minimal volume sections by symmetric slabs of a large fixed width. This parallels recent results about noncentral sections of the cube due to Moody, Stone, Zach and Zvavitch.
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 $n > k \geqslant 2.$
Optimal Extensions of Cross-Sections: Sphere Packings in Dimensions 38 to 43
State Key Lab of Processors, Institute of Computing Technology, Chinese Academy of Sciences（处理器重点实验室，中国科学院计算技术研究所）；University of Chinese Academy of Sciences（中国科学院大学）
We establish the log-concavity of the volume of central sections of dilations of the cross-polytope (the strong B-inequality for the cross-polytope and Lebesgue measure restricted to an arbitrary subspace).
Although previous research has found several facts concerning chord lengths of regular polytopes, none of these investigations has considered whether any of these facts define relationships that might generalize to the chord lengths of all regular polytopes. Consequently, this paper explores whether four findings of previous studies-- viz., the four facts relating to the sums and products of squared chord lengths of regular polygons inscribed in unit circles-- can be generalized to all regular ($n$-dimensional) polytopes (inscribed in unit $n$-spheres). We show that (a) one of these four facts…
Let C = C(l_1, ..., l_n) be the n-dimensional orthogonal cross-polytope whose axes are of length l_1,..., l_n. Subject to the condition \sum l_i^2 = 1, the mean width of C is minimised when l_i = 1/sqrt{n} for every i, and it is maximised when C is at most two dimensional. As a corollary, a lower bound on the mean width of a general convex body K is derived in terms of the successive inner radii of K. A more general result is presented for Gaussian random vectors.
The expected range of a sample of n+1 normally distributed variables is known to be related to the mean width of a regular n-simplex. We show that the expected maximum mu_n of a sample of n half-normally distributed variables is related to the mean width of a regular n-crosspolytope. Both of these relations have mean square counterparts. An expression for mu_5 is found and is believed to be new.
Let $n\geq 3$, and let $B_1^n$ be the standard $n$-dimensional cross-polytope (i.e. the convex hull of standard coordinate vectors and their negatives). We show that there exists a symmetric convex body $\mathcal G_m$ in ${\mathbb R}^n$ such that the Banach--Mazur distance $d(B_1^n,\mathcal G_m)$ satisfies $d(B_1^n,\mathcal G_m)\geq n^{5/9}\log^{-C}n$, where $C>0$ is a universal constant. The body $\mathcal G_m$ is obtained as a typical realization of a random polytope in ${\mathbb R}^n$ with $2m:=2n^C$ vertices (for a large constant $C$). The result improves upon an earlier estimate of S.Szar…
We study three families of polyhedral cones whose sections are regular simplices, cubes, and crosspolytopes. We compute solid angles and conic intrinsic volumes of these cones. We show that several quantities appearing in stochastic geometry can be expressed through these conic intrinsic volumes. A list of such quantities includes internal and external solid angles of regular simplices and crosspolytopes, the probability that a (symmetric) Gaussian random polytope or the Gaussian zonotope contains a given point, the expected number of faces of the intersection of a regular polytope with a rand…
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.
Wythoff's construction associates a uniform polytope to a Coxeter diagram whose vertices are decorated with crosses, which indicate the subgroup stabilizing a generic point. Champagne, Kjiri, Patera, and Sharp remarked that by associating more information to a Coxeter diagram, one can furthermore determine the types and number of faces of such a polytope. This article provides a proof of this result, and uses it to provide a classification of the regular polytopes.
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…
Let $X_1,\ldots,X_n$ be a standard normal sample in $\mathbb R^d$. We compute exactly the expected volume of the Gaussian polytope $\mathrm{conv}[X_1,\ldots,X_n]$, the symmetric Gaussian polytope $\mathrm{conv}[\pm X_1,\ldots,\pm X_n]$, and the Gaussian zonotope $[0,X_1]+\ldots+[0,X_n]$ by exploiting their connection to the regular simplex, the regular crosspolytope, and the cube with the aid of Tsirelson's formula. The expected volumes of these random polytopes are given by essentially the same expressions as the intrinsic volumes and external angles of the regular polytopes. For all these qu…
We present higher dimensional versions of the classical results of Euler and Fuss, both of which are special cases of the celebrated Poncelet porism. Our results concern polytopes, specifically simplices, parallelotopes and cross polytopes, inscribed in a given ellipsoid and circumscribed to another. The statements and proofs use the language of linear algebra. Without loss, one of the ellipsoids is the unit sphere and the other one is also centered at the origin. Let $A$ be the positive symmetric matrix taking the outer ellipsoid to the inner one. If $trace A = 1$, there exists a bijection be…
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…
An old conjecture states that among all simplices inscribed in the unit sphere the regular one has the maximal mean width. An equivalent formulation is that for any centered Gaussian vector $(ξ_1,\dots,ξ_n)$ satisfying $\mathbb Eξ_1^2= \dots =\mathbb Eξ_n^2=1$ one has $$ \mathbb E\,\max\{ξ_1,\dots,ξ_n\}\leq\sqrt{\frac{n}{n-1}}\, \mathbb E\,\max\{η_1,\dots,η_n\}, $$ where $η_1,η_2,\dots,$ are independent standard Gaussian variables. Using this probabilistic interpretation we derive an asymptotic version of the conjecture. We also show that the mean width of the regular simplex with $2n$ vertice…
