For $N\geq n$, let $P_{ N,n}$ be a random polytope in ${ \mathbb R}^n$ with vertices $\pm X_i$, $1\leq i\leq N$, where $X_1,\dots,X_N$ are i.i.d standard Gaussian vectors in ${ \mathbb R}^n$. Random polytopes $P_{ N,n}$, as well as their duals, are classical objects of interest in high-dimensional convex geometry and local Banach space theory. In this paper, we provide a { \it dimension-independent} bound on the cotype of the corresponding normed space $({ \mathbb R}^n,\|\cdot\|_{ P_{ N,n}})$, generated by $P_{ N,n}$. Let $K'\geq K>1$, and assume that $K'\geq \frac{ N}{ n}\geq K$. We show that with probability $1-o(1)$, for any $k\geq 1$, and any collection $y_1,\dots,y_k$ of vectors in ${ \mathbb R}^n$, $$ { \mathbb E}_\sigma\,\Big\|\sum_{ i=1}^k \sigma_i y_i\Big\|_{ P_{ N,n}}^q \geq \frac{ 1}{ C_q^q}\sum_{ i=1}^k \big\|y_i\big\|_{ P_{ N,n}}^q, $$ where $\sigma=(\sigma_1,\dots,\sigma_k)$ is a vector of random signs, and where $q\in [2,\infty)$ and $C_q\in[1,\infty)$ may only depend on $K,K'$. We discuss the result in context of infinite-dimensional Banach spaces.
Categories: math.FA math.MG math.PR