Dimension-free cotype for isotropic log-concave random polytope spaces

Abstract

Let $X_1,\ldots,X_N$ be independent random vectors in $\mathbb{R}^n$ with common isotropic log-concave distribution $μ$ and set $P_{N,n}^μ:=\operatorname{conv}\{\pm X_i:1\leqslant i\leqslant N\}$. Assume that $N/n=γ\geqslant γ_0$ where $γ_0>1$ is an absolute constant. We prove that with probability at least $1-Cγ\exp(-c n^{1/4})$ every $k$-dimensional subspace $E$ of $(\mathbb{R}^n,\|\cdot\|_{P_{N,n}^μ})$ satisfies $d_{\mathrm{BM}} (E,\ell_\infty^k) \geqslant cγ^{-C}k^α$ for every $1\leqslant k\leqslant n$ where $c,C,α>0$ are absolute constants. Consequently, with the same probability, $(\mathbb{R}^n,\|\cdot\|_{P_{N,n}^μ})$ has cotype $q(γ)<\infty$ with cotype constant depending only on $γ$, in particular the cotype exponent and the cotype constant are independent of $n$ and of $μ$. The proof adapts the deterministic coefficient scheme of Huang-Tikhomirov replacing the Gaussian estimates in their argument by estimates for isotropic log-concave random matrices. As an application, using the log-concave extension of Gluskin's theorem, we obtain a separable Banach space of finite cotype for which the Banach-Mazur diameter of its $k$-dimensional subspaces is of order $k$ and whose finite-dimensional building blocks are generated by isotropic log-concave random polytopes.