Optimal $M{M}^{\ast }$ bounds for convex bodies

Abstract

Let $K\subset {\mathbb{R}}^n$ be a convex body in isotropic position. We prove the optimal mean-width estimate \[ M^*(K)\leq C\sqrt{n\log n}. \] The main new ingredient is a geometric inequality relating the Gaussian mean of the support function to its mean under the uniform measure on $K$, obtained through a heat-flow argument. Combined with the Gaussian-log-concave comparison of Eldan and Lehec and the newly available dimension-free bound on the third-moment parameter $\hat{I}{º}_n$, this yields the result. The boundedness of $\hat{I}{º}_n$ also makes the mean-norm estimate of Bizeul and Klartag sharp. Combining both estimates yields \[ M(K)M^*(K)\leq C\log n, \] extending Pisier's $M{M}^{\ast }$ estimate to non-symmetric convex bodies and to the isotropic position.

20 pages