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arXiv · 2507.12099

On p-Brunn-Minkowski and Brascamp-Lieb inequalities

Alexander V. Kolesnikov·Galyna Livshyts·Liran Rotem
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Abstract

We show that a strong version of the Brascamp--Lieb inequality for symmetric log-concave measure with $\alpha$-homogeneous potential $V$ is equivalent to a $p$-Brunn--Minkowski inequality for level sets of $V$ with some $p(\alpha,n)<0$. We establish links between several inequalities of this type on the sphere and the Euclidean space. Exploiting these observations, we prove new sufficient conditions for symmetric $p$-Brunn--Minkowski inequality with $p<1$. In particular, we prove the local log-Brunn--Minkowski for $L_q$-balls for all $q\geq 1$ in all dimensions, which was previously known only for $q\geq 2$.

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Alexander V. Kolesnikov, Galyna Livshyts, Liran Rotem. 2025-07-16. On p-Brunn-Minkowski and Brascamp-Lieb inequalities. https://arxiv.org/abs/2507.12099

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