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Perfect and almost perfect homogeneous polytopes

V. N. Berestovskii, Yu. G. Nikonorov

Abstract

The paper is devoted to perfect and almost perfect homogeneous polytopes in Euclidean spaces. We classified perfect and almost perfect polytopes among all regular polytopes and all semiregular polytopes excepting Archimedean solids and two four-dimensional Gosset polytopes. Also we construct some non-regular homogeneous polytopes that are (or are not) perfect and posed some unsolved questions.

Perfect and almost perfect homogeneous polytopes

Abstract

The paper is devoted to perfect and almost perfect homogeneous polytopes in Euclidean spaces. We classified perfect and almost perfect polytopes among all regular polytopes and all semiregular polytopes excepting Archimedean solids and two four-dimensional Gosset polytopes. Also we construct some non-regular homogeneous polytopes that are (or are not) perfect and posed some unsolved questions.
Paper Structure (4 sections, 24 theorems, 17 equations, 1 figure)

This paper contains 4 sections, 24 theorems, 17 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Some natural examples
  3. The reducibility, Löwner --- John ellipsoids, and the almost perfectness
  4. Examples of perfect polytopes

Key Result

Proposition 1

Let $M=\{x_1, \dots, x_q\}$, $q\geq n+1$, be a finite homogeneous metric subspace of Euclidean space $\mathbb{R}^n$, $n\geq 2$. Then $M$ is the vertex set of a convex polytope $P$, that is situated in some sphere in $\mathbb{R}^n$ with radius $r>0$ and center $x_0=\frac{1}{q}\cdot\sum_{k=1}^{q}x_k$.

Figures (1)

  • Figure 1: Illustrations for: a) Example \ref{['ex:dodec1']}; b) Proposition \ref{['pr.icosa1']}.

Theorems & Definitions (43)

  • Proposition 1: BerNik19
  • Definition 1
  • Definition 2
  • Lemma 1: BerNik21n
  • Lemma 2
  • Corollary 1
  • Theorem 1
  • Lemma 3
  • Proposition 2
  • Example 1
  • ...and 33 more