Distances between non-symmetric convex bodies: optimal bounds up to polylog
Pierre Bizeul, Boaz Klartag
TL;DR
This paper proves an improved polylog-enhanced bound on the non-symmetric Banach-Mazur distance: for any convex bodies $K_1,K_2\subset\mathbb{R}^n$, $d_{BM}(K_1,K_2)\le C n\,\log^{\alpha}(n+1)$ with universal constants $C,\alpha>0$, achieving $\alpha\le 4$ (and effectively optimal up to polylog factors). It further shows a similar polylog bound for the partial containment distance $d_{PC}(K_1,K_2)\le C \log^{\alpha}(n+1)$, with the block that isotropic random rotations realize these bounds. The approach couples the $M M^*$ method with stochastic localization to control the isotropic bounds $M(K)$ and $M^*(K)$ via Gaussian comparisons and to transfer them to operator norms through Chevet-type inequalities. The results align the non-symmetric case with the classical symmetric bounds up to polylog factors and identify random isotropic position as the attaining regime.
Abstract
We show that the non-symmetric Banach-Mazur distance between two convex bodies $K_1, K_2 \subseteq \mathbb{R}^n$ satisfies $$ d_{BM}(K_1, K_2) \leq C n \cdot \log^α (n+1), $$ for universal constants $C, α> 0$. This improves upon the earlier bound $C n^{4/3} \log^α (n+1)$ due to Rudelson. Up to polylogarithmic factors, our estimate is optimal and it also matches the optimal bound in the centrally-symmetric case which is realized in the John position, as proven by Gluskin. The bound above for the Banach-Mazur distance is attained when both bodies are in a ``random isotropic position'', that is, in isotropic position after a random rotation. Our proof is based on an $M$-bound in the isotropic position, which complements E. Milman's $M^*$-bound. In addition, we consider the partial containment distance $d_{PC}(K_1, K_2)$ between two convex bodies $K_1, K_2 \subseteq \mathbb{R}^n$, where the Banach-Mazur requirement to contain $100\%$ of the other body is relaxed to $99\%$-containment. We prove that for any pair of convex bodies $K_1, K_2 \subseteq \mathbb{R}^n$, $$ d_{PC}(K_1, K_2) \leq C \log^α (n+1), $$ and that any isotropic position of $K_1$ and $K_2$ yields this polylogarithmic bound for $d_{PC}$.