On the geometry of random polytopes

Abstract

We present a simple proof to a fact recently established in [5]: let $\xi$ be a symmetric random variable that has variance $1$, let $\Gamma =({\xi }_{ij})$ be an $N\times n$ random matrix whose entries are independent copies of $\xi$, and set $X_1,...,X_N$ to be the rows of $\Gamma$. Then under minimal assumptions on $\xi$ and as long as $N\ge c_{1}n$, $$c_2(B_{\infty }^n\cap \sqrt{\log (eN/n)}{B}_2^n)\subset \mathrm{absconv}(X_1,...,X_N)$$ with high probability.