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On the Many Faces of Easily Covered Polytopes

Dan I. Florentin, Tomer Milo

Abstract

Assume that $rB_{2}^{n} \subset P$ for some polytope $P \subset \mathbb{R}^n$, where $r \in (\frac{1}{2},1]$. Denote by $\mathcal{F}$ the set of facets of $P$, and by $N=N(P,B_2^n)$ the covering number of $P$ by the Euclidean unit ball $B_2^n$. We prove that if $\log N \le\frac{n}{8}$, then \[ |\mathcal{F}| \ge \left( \frac{1}{ 2\left(1 - r \sqrt{1-\frac{4\log N}{n}}\right) } \right)^{\frac{n-1}{2}}. \]

On the Many Faces of Easily Covered Polytopes

Abstract

Assume that for some polytope , where . Denote by the set of facets of , and by the covering number of by the Euclidean unit ball . We prove that if , then

Paper Structure

This paper contains 2 sections, 6 theorems, 30 equations.

Table of Contents

  1. Introduction and the main result
  2. Proof of Theorem \ref{['thm: main theorem']}

Key Result

Theorem 1.1

Let $n\ge 3$ and let $P\subset \mathbb R^n$ be a polytope containing $rB_2^n$, for some $\frac{3\sqrt{3}}{\sqrt{n}}\le r \le 1$. Then where $\mathcal{F}$ is the set of facets of $P$.

Theorems & Definitions (13)

  • Conjecture : Milman-Szarek
  • Theorem 1.1
  • Proposition 1.2
  • proof
  • Proposition 2.1
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Proposition 2.4
  • proof
  • ...and 3 more