On p-Brunn-Minkowski and Brascamp-Lieb inequalities

TL;DR

The paper builds a comprehensive bridge between strong Brascamp–Lieb inequalities for symmetric log-concave measures with $\alpha$-homogeneous potentials and local $p$-Brunn–Minkowski inequalities for level sets of the associated radial function. It develops a dual framework on the sphere, converts BL bounds to dual BL-type inequalities via Legendre transforms, and derives explicit spectral-gap constants that yield local $p$-BM results, including local log-BM for $L^q$ balls in all dimensions. By exploiting Bochner formulas and decomposition into parity components, it establishes pinching estimates and unconditional-bodies criteria, leading to concrete BM constants and Gaussian-characterization results. The work further unifies these inequalities with a Hessian-geometry view and revisits Blaschke–Santaló-type inequalities, showing that BL-type inequalities imply local BM forms and that, in Gaussian (2-homogeneous) cases, the level sets must be ellipsoids. Overall, the results advance a spectral-geometry approach to convex-geometric inequalities with broad implications for analysis on Hessian manifolds and optimal transport.

Abstract

We show that a strong version of the Brascamp--Lieb inequality for symmetric log-concave measure with $α$-homogeneous potential $V$ is equivalent to a $p$-Brunn--Minkowski inequality for level sets of $V$ with some $p(α,n)<0$. 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 $q\geq 2$.

Theorems & Definitions (55)

• Remark 1.1
• Definition 1.2
• Remark 1.3
• Theorem 1.4
• Corollary 1.5
• Corollary 1.6
• Remark 1.7
• Example 1.8
• Theorem 1.9
• Theorem 1.10