Cotype of random polytopes

Abstract: For $N\geq n$, let $P_{N,n}$ be a random polytope in ${\mathbb R}<sup>n$ with vertices $\pm X_i$, $1\leq i\leq N$, where $X_1,\dots,X_N$ are i.i.d standard Gaussian vectors in ${\mathbb R}<sup>n$. 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 $({\mathbb R}<sup>n,|\cdot|<em>{P</em>{N,n}})$, generated by $P_{N,n}$. Let $K&#39;\geq K&gt;1$, and assume that $K&#39;\geq \frac{N}{n}\geq K$. We show that with probability $1-o(1)$, for any $k\geq 1$, and any collection $y_1,\dots,y_k$ of vectors in ${\mathbb R}<sup>n$, $${\mathbb E}<em>σ\,\Big|\sum</em>{i=1}^k σ<em>i y_i\Big|</em>{P_{N,n}}^q \geq \frac{1}{C_q^{q}\sum_{i=1}^k} \big|y_i\big|<em>{P</em>{N,n}}^q,$$ where $σ=(σ_1,\dots,σ_k)$ is a vector of random signs, and where $q\in [2,\infty)$ and $C_q\in[1,\infty)$ may only depend on $K,K&#39;$. We discuss the result in context of infinite-dimensional Banach spaces.