Related papers: Radii of regular polytopes
There exists just one regular polytope of rank larger than 3 whose full automorphism group is a projective general linear group PGL_2(q), for some prime-power q. This polytope is the 4-simplex and the corresponding group is PGL_2(5), which…
We prove inequalities involving intrinsic and extrinsic radii and diameters of tetrahedra.
Given a fixed $p\neq 2$, we prove a simple and effective characterization of all radial multipliers of $\cF L^p(\Bbb R^d)$, provided that the dimension $d$ is sufficiently large. The method also yields new $L^q$ space-time regularity…
In the classical setting, a convex polytope is said to be semiregular if its facets are regular and its symmetry group is transitive on vertices. This paper studies semiregular abstract polytopes, which have abstract regular facets, still…
This paper presents a sharp upper bound for the spectral radius of simple digraphs with described number of arcs. Further, the extremal graphs which attain the maximum spectral radius among all simple digraphs with fixed arcs are…
We discuss new limits on masses and radii of compact stars and we conclude that they can be interpreted as an indication of the existence of two classes of stars: "normal" compact stars and "ultra-compact" stars. We estimate the critical…
The existence, radii and radial stability of the equatorial and non-equatorial (particularly, the polar) spherical orbits are discussed for particles with different conserved energy. The radii of these orbits generally are solutions of a…
A compact circle-packing $P$ of the Euclidean plane is a set of circles which bound mutually disjoint open discs with the property that, for every circle $S\in P$, there exists a maximal indexed set $\{A_{0},\ldots,A_{n-1}\}\subseteq P$ so…
Abstract polytopes are a combinatorial generalization of convex and skeletal polytopes. Counting how many flag orbits a polytope has under its automorphism group is a way of measuring how symmetric it is. Polytopes with one flag orbit are…
We study Forman--Ricci and effective resistance curvatures on the skeleta of convex polytopes. Our guiding questions are: how frequently do polytopal graphs exhibit everywhere positive curvature, and what structural constraints does…
The binding energies, shapes and sizes of even-even beta-stable nuclei with A >= 40 and a few chains of isotopes with Z=50, 56, 82, 94 protons and isotones with N=50, 82, 126 neutrons are analyzed. The average isospin dependence of the…
The conjecture of Kalai, Kleinschmidt, and Lee on the number of empty simplices of a simplicial polytope is established by relating it to the first graded Betti numbers of the polytope. The proof allows us to derive explicit optimal bounds…
This paper discusses properties of the graphs of 2-way and 3-way transportation polytopes, in particular, their possible numbers of vertices and their diameters. Our main results include a quadratic bound on the diameter of axial 3-way…
If we fix the angles at the vertices of a convex planar $n$-gon, the lengths of its edges must satisfy two linear constraints in order for it to close up. If we also require unit perimeter, our vectors of $n$ edge lengths form a convex…
Persistent homology has emerged as a novel tool for data analysis in the past two decades. However, there are still very few shapes or even manifolds whose persistent homology barcodes (say of the Vietoris-Rips complex) are fully known.…
This work is a survey on completely regular codes. Known properties, relations with other combinatorial structures and constructions are stated. The existence problem is also discussed and known results for some particular cases are…
Marginal polytopes are important geometric objects that arise in statistics as the polytopes underlying hierarchical log-linear models. These polytopes can be used to answer geometric questions about these models, such as determining the…
This article presents some interesting and novel results concerning the average modulus of random polynomials on the unit circle and the unit disc, with coefficients distributed as standard normal variates. The paper also introduces new…
The collection $\mathcal{M}_n$ of all metric spaces on $n$ points whose diameter is at most $2$ can naturally be viewed as a compact convex subset of $\mathbb{R}^{\binom{n}{2}}$, known as the metric polytope. In this paper, we study the…
It is known that polytopes with at most two nonsimple vertices are reconstructible from their graphs, and that $d$-polytopes with at most $d-2$ nonsimple vertices are reconstructible from their 2-skeletons. Here we close the gap between 2…