On the geometry of random polytopes

Abstract

We present a simple proof to a fact recently established in [5]: let $ξ$ be a symmetric random variable that has variance $1$, let $Γ=(ξ_{ij})$ be an $N \times n$ random matrix whose entries are independent copies of $ξ$, and set $X_1,...,X_N$ to be the rows of $Γ$. Then under minimal assumptions on $ξ$ and as long as $N \geq c_1n$, $$ c_2 \bigl(B_\infty^n \cap \sqrt{\log(eN/n)} B_2^n \bigr) \subset {\rm absconv}(X_1,...,X_N) $$ with high probability.