Cotype of random polytopes

Abstract

For $N\ge n$, let $P_{N,n}$ be a random polytope in ${\mathbb{R}}^n$ with vertices $\pm X_i$, $1\le i\le N$, 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 $({\mathbb{R}}^n,\Vert \cdot {\Vert }_{P_{N,n}})$, generated by $P_{N,n}$. Let $K^{\prime }\ge K>1$, and assume that $K^{\prime }\ge \frac{N}{n}\ge K$. We show that with probability $1-o(1)$, for any $k\ge 1$, and any collection $y_1,\dots ,y_k$ of vectors in ${\mathbb{R}}^n$, $${\mathbb{E}}_{\ddot{I}}ƒ\Vert \sum _{i=1}^k\ddot{I}{ƒ}_i{y}_i{\Vert }_{P_{N,n}}^q\ge \frac{1}{C_q^q}\sum _{i=1}^k\Vert y_i{\Vert }_{P_{N,n}}^q,$$ where $\ddot{I}ƒ=(\ddot{I}{ƒ}_1,\dots ,\ddot{I}{ƒ}_k)$ is a vector of random signs, and where $q\in [2,\infty )$ and $C_q\in [1,\infty )$ may only depend on $K,K^{\prime }$. We discuss the result in context of infinite-dimensional Banach spaces.