Optimal $MM^*$ bounds for convex bodies

Pierre Bizeul

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arXiv · 2607.29458

Optimal $MM^*$ bounds for convex bodies

Pierre Bizeul

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Abstract

Let $K\subset\R^n$ 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 $\kappa_n$, this yields the result. The boundedness of $\kappa_n$ also makes the mean-norm estimate of Bizeul and Klartag sharp. Combining both estimates yields \[ M(K)M^*(K)\leq C\log n, \] extending Pisier's $MM^*$ estimate to non-symmetric convex bodies and to the isotropic position.

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Pierre Bizeul. 2026-07-31. Optimal $MM^*$ bounds for convex bodies. https://arxiv.org/abs/2607.29458

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