A universal bound in the dimensional Brunn-Minkowski inequality for log-concave measures

Galyna V. Livshyts

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arXiv · 2107.00095

A universal bound in the dimensional Brunn-Minkowski inequality for log-concave measures

Galyna V. Livshyts

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Abstract

We show that for any log-concave measure $\mu$ on $\mathbb{R}^n$, any pair of symmetric convex sets $K$ and $L$, and any $\lambda\in [0,1],$ $$\mu((1-\lambda) K+\lambda L)^{c_n}\geq (1-\lambda) \mu(K)^{c_n}+\lambda\mu(L)^{c_n},$$ where $c_n\geq n^{-4-o(1)}.$ This constitutes progress towards the dimensional Brunn-Minkowski conjecture (see Gardner, Zvavitch \cite{GZ}, Colesanti, L, Marsiglietti \cite{CLM}). Moreover, our bound improves for various special classes of log-concave measures.

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Galyna V. Livshyts. 2021-06-30. A universal bound in the dimensional Brunn-Minkowski inequality for log-concave measures. https://arxiv.org/abs/2107.00095

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