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arXiv · 2311.05446

New Brunn--Minkowski and functional inequalities via convexity of entropy

Gautam Aishwarya·Liran Rotem
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Abstract

We study the connection between the concavity properties of a measure $\nu$ and the convexity properties of the associated relative entropy $D(\cdot \Vert \nu)$ along optimal transport. As a corollary we prove a new dimensional Brunn--Minkowski inequality for centered star-shaped bodies, when the measure $\nu$ is log-concave with a p-homogeneous potential (such as the Gaussian measure). Our method allows us to go beyond the usual convexity assumption on the sets that is fundamentally essential for the standard differential-geometric technique in this area. We then take a finer look at the convexity properties of the Gaussian relative entropy, which yields new functional inequalities. First we obtain curvature and dimensional reinforcements to Otto--Villani's HWI inequality in Gauss space, when restricted to even strongly log-concave measures. As corollaries, we obtain improved versions of Gross' Logarithmic Sobolev inequality and Talagrand's transportation cost inequality in this setting.

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BibTeXRIS

Gautam Aishwarya, Liran Rotem. 2023-11-09. New Brunn--Minkowski and functional inequalities via convexity of entropy. https://doi.org/10.1016/j.aim.2026.110841

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