Distances between non-symmetric convex bodies: optimal bounds up to polylog

We show that the non-symmetric Banach-Mazur distance between two convex bodies $K_1,K_2\subseteq {\mathbb{R}}^n$ satisfies $$d_{BM}(K_1,K_2)\le Cn\cdot {\log }^{\alpha }(n+1),$$ for universal constants $C,\alpha >0$. This improves upon the earlier bound $C{n}^{4\mathrm{/}3}{\log }^{\alpha }(n+1)$ 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 $M^{\ast }$-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 $100\mathrm{\%}$ of the other body is relaxed to $99\mathrm{\%}$-containment. We prove that for any pair of convex bodies $K_1,K_2\subseteq {\mathbb{R}}^n$, $$d_{PC}(K_1,K_2)\le C{\log }^{\alpha }(n+1),$$ and that any isotropic position of $K_1$ and $K_2$ yields this polylogarithmic bound for $d_{PC}$.