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

Let $X_1,\dots ,X_N$ be independent random vectors in ${\mathbb{R}}^n$ with common isotropic log-concave distribution $\mu$ and set $P_{N,n}^{\mu }:=\mathrm{conv}\{\pm X_i:1⩽i⩽N\}$. Assume that $N\mathrm{/}n=\gamma ⩾{\gamma }_0$ where ${\gamma }_0>1$ is an absolute constant. We prove that with probability at least $1-C\gamma \exp (-c{n}^{1\mathrm{/}4})$ every $k$-dimensional subspace $E$ of $({\mathbb{R}}^n,\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{P_{N,n}^{\mu }})$ satisfies $d_{\mathrm{B}\mathrm{M}}(E,{\mathrm{\ell }}_{\mathrm{\infty }}^k)⩾c{\gamma }^{-C}{k}^{\alpha }$ for every $1⩽k⩽n$ where $c,C,\alpha >0$ are absolute constants. Consequently, with the same probability, $({\mathbb{R}}^n,\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{P_{N,n}^{\mu }})$ has cotype $q(\gamma )<\mathrm{\infty }$ 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.