On the Many Faces of Easily Covered Polytopes

TL;DR

This paper investigates the geometric properties of polytopes that can be easily covered by Euclidean balls, establishing bounds on the number of facets based on covering numbers and containment conditions.

Contribution

It provides a new lower bound on the number of facets of polytopes that are easily covered by Euclidean balls, linking covering numbers and facet counts.

Findings

Derived a lower bound on the number of facets based on covering number constraints.

Established a relationship between polytope containment and facet complexity.

Extended understanding of geometric covering properties of polytopes.

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 \[ |\mathcal{F}| \ge \left( \frac{1}{ 2\left(1 - r \sqrt{1-\frac{4\log N}{n}}\right) } \right)^{\frac{n-1}{2}}. \]