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

Livshyts, Galyna

doi:10.1090/tran/8976

A universal bound in the dimensional Brunn-Minkowski inequality for log-concave measures
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by Galyna V. Livshyts;

Trans. Amer. Math. Soc. 376 (2023), 6663-6680

DOI: https://doi.org/10.1090/tran/8976

Published electronically: June 16, 2023

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Abstract:

We show that for any even log-concave probability measure $\mu$ on $\mathbb {R}^n$, any pair of symmetric convex sets $K$ and $L$, and any $\lambda \in [0,1]$, \begin{equation*} \mu ((1-\lambda ) K+\lambda L)^{c_n}\geq (1-\lambda ) \mu (K)^{c_n}+\lambda \mu (L)^{c_n}, \end{equation*} where $c_n\geq n^{-4-o(1)}$. This constitutes progress towards the dimensional Brunn-Minkowski conjecture (see Richard J. Gardner and Artem Zvavitch [Tran. Amer. Math. Soc. 362 (2010), pp. 5333–5353]; Andrea Colesanti, Galyna V. Livshyts, Arnaud Marsiglietti [J. Funct. Anal. 273 (2017), pp. 1120–1139]). Moreover, our bound improves for various special classes of log-concave measures.

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