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

Let $X_1,\ldots,X_N$ be independent random vectors in $\mathbb{R}^n$ with common isotropic log-concave distribution $\mu$ and set $P_{N,n}^{\mu}:=\operatorname{conv}\{\pm X_i:1\leqslant i\leqslant N\}$. Assume that $N/n=\gamma\geqslant \gamma_0$ where $\gamma_0>1$ is an absolute constant. We prove that with probability at least $1-C\gamma\exp(-c n^{1/4})$ every $k$-dimensional subspace $E$ of $(\mathbb{R}^n,\|\cdot\|_{P_{N,n}^{\mu}})$ satisfies $d_{\mathrm{BM}} (E,\ell_\infty^k) \geqslant c\gamma^{-C}k^\alpha$ for every $1\leqslant k\leqslant n$ where $c,C,\alpha>0$ are absolute constants. Consequently, with the same probability, $(\mathbb{R}^n,\|\cdot\|_{P_{N,n}^{\mu}})$ has cotype $q(\gamma)<\infty$ with cotype constant depending only on $\gamma$, in particular the cotype exponent and the cotype constant are independent of $n$ and of $\mu$. 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.