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 $$d_{BM}(K_1, K_2) \leq C n \cdot \log^{\alpha} (n+1),$$ for universal constants $C, \alpha > 0$. This improves upon the earlier bound $C n^{4/3} \log^{\alpha} (n+1)$ 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 $M^*$-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 $K_1, K_2 \subseteq \mathbb{R}^n$, $$d_{PC}(K_1, K_2) \leq C \log^{\alpha} (n+1),$$ and that any isotropic position of $K_1$ and $K_2$ yields this polylogarithmic bound for $d_{PC}$.