Optimal MM* bounds for convex bodies

Let $K\subset {\mathbb{R}}^n$ be a convex body in isotropic position. We prove the optimal mean-width estimate $$M^{\ast }(K)\le C\sqrt{n\log n}\mathrm{.}$$ 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 ${\kappa }_n$, this yields the result. The boundedness of ${\kappa }_n$ also makes the mean-norm estimate of Bizeul and Klartag sharp. Combining both estimates yields $$M(K)M^{\ast }(K)\le C\log n,$$ extending Pisier’s $M{M}^{\ast }$ estimate to non-symmetric convex bodies and to the isotropic position.