Banach-Mazur distances and basis constants of isotropic log-concave random spaces

Abstract

We study the Banach-Mazur distance between random normed spaces generated by centrally symmetric random polytopes associated with isotropic log-concave measures in $\mathbb{R}^n$. We show that, in a wide range of parameters, if $x_1,\dots,x_m$ and $y_1,\dots,y_m$ are independent samples from an isotropic log-concave probability measure on $\mathbb{R}^n$, then the corresponding normed spaces $X_{B_m}$ and $Y_{A_m}$ generated by their absolute convex hulls satisfy, with high probability, $$d_{{\rm BM}}(X_{B_m},Y_{A_m}) \geqslant \frac{cn}{\ln(1+m/n)},$$ which is sharp in both $n$ and $m$ and recovers the extremal order $n$ when $m \approx n$. Our results extend Gluskin's theorem from the Gaussian setting to general isotropic log-concave measures, providing evidence for a universality phenomenon in the extremal geometry of the Banach-Mazur compactum. In addition, we investigate operator-theoretic properties of the associated random spaces and, as consequences, we derive sharp estimates for their basis constant and show that these random spaces are far from the class of spaces with a $1$-unconditional basis. The proofs combine probabilistic and geometric methods with recent advances related to Bourgain's slicing problem.