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

Abstract

Let $X_1,\dots ,X_N$ be independent random vectors in ${\mathbb{R}}^n$ with common isotropic log-concave distribution $\hat{I}¼$ and set $P_{N,n}^{\hat{I}}¼:=\mathrm{conv}\{\pm X_i:1⩽i⩽N\}$. Assume that $N/n={\hat{I}}^3⩾{\hat{I}}_0^3$ where ${\hat{I}}_0^3>1$ is an absolute constant. We prove that with probability at least $1-C{\hat{I}}^3\exp (-c{n}^{1/4})$ every $k$-dimensional subspace $E$ of $({\mathbb{R}}^n,\Vert \cdot {\Vert }_{P_{N,n}^{\hat{I}}¼})$ satisfies $d_{\mathrm{BM}} (E,\ell_\infty^k) \geqslant cγ^{-C}k^α$ for every $1⩽k⩽n$ where $c,C,\hat{I}\pm >0$ are absolute constants. Consequently, with the same probability, $({\mathbb{R}}^n,\Vert \cdot {\Vert }_{P_{N,n}^{\hat{I}}¼})$ has cotype $q({\hat{I}}^3)<\infty$ with cotype constant depending only on ${\hat{I}}^3$, in particular the cotype exponent and the cotype constant are independent of $n$ and of $\hat{I}¼$. 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.