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

In this paper we determine, up to polylogarithmic factors, the diameter of the Banach--Mazur compactum of $n$-dimensional convex bodies without symmetry assumptions. We prove that for any convex bodies $K_1,K_2\subset \mathbb{R}^n$, \begin{equation} d_{BM}(K_1,K_2)\le Cn\log^\alpha(n+1), \label{eq_1624} \end{equation} for universal constants $C,\alpha>0$, improving an earlier bound of Rudelson. We also study the partial-containment distance $d_{PC}$, in which the Banach-Mazur requirement to contain the other body in its entirety is relaxed to $99\%$-containment. We prove that this relaxation leads to a very different behavior: \begin{equation} d_{PC}(K_1,K_2) \leq C \log^{\alpha} (n+1) \label{eq_1625} \end{equation} for all convex bodies $K_1,K_2 \subseteq \mathbb{R}^n$. This demonstrates that in high dimensions, any convex body is not too far from an affine image of any other convex body, when we look at the bulk of their mass.
\medskip In the centrally-symmetric case, the optimal upper bound for the Banach-Mazur distance is obtained in the John position. In contrast, our proofs rely on the isotropic position. The analytic core of our argument is a two-sided comparison, in the gauge order, between isotropic log-concave measures and Gaussian measures. This yields a new isotropic $M$-bound that complements E. Milman's $M^*$-bound. We also provide applications to linear symplectic geometry and to the first Dirichlet eigenvalue of the Laplacian.