On the edge expansion of random polytopes

Abstract

A $0/1$-polytope in $\mathbb{R}^n$ is the convex hull of a subset of $\{0,1\}^n$. The graph of a polytope $P$ is the graph whose vertices are the zero-dimensional faces of $P$ and whose edges are the one-dimensional faces of $P$. A conjecture of Mihail and Vazirani states that the edge expansion of the graph of every $0/1$-polytope is at least one. We study a random version of the problem, where the polytope is generated by selecting vertices of $\{0,1\}^n$ independently at random with probability $p\in (0,1)$. Improving earlier results, we show that, for any $p\in (0,1)$, with high probability the edge expansion of the random $0/1$-polytope is bounded from below by an absolute constant.