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

Gautam Aishwarya, Liran Rotem

Abstract

We study the connection between the concavity properties of a measure $ν$ and the convexity properties of the associated relative entropy $D(\cdot \Vert ν)$ along optimal transport. As a corollary we prove a new dimensional Brunn-Minkowski inequality for centered star-shaped bodies, when the measure $ν$ 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 the Gauss space, when restricted to even strongly log-concave measures. As corollaries, we obtain improved versions of Gross' logarithmic Sobolev inequality and Talgrand's transportation cost inequality in this setting.

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

Abstract

We study the connection between the concavity properties of a measure and the convexity properties of the associated relative entropy along optimal transport. As a corollary we prove a new dimensional Brunn-Minkowski inequality for centered star-shaped bodies, when the measure 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 the Gauss space, when restricted to even strongly log-concave measures. As corollaries, we obtain improved versions of Gross' logarithmic Sobolev inequality and Talgrand's transportation cost inequality in this setting.
Paper Structure (7 sections, 27 theorems, 77 equations)

This paper contains 7 sections, 27 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Acknowledgement:
  3. Declarations of interest
  4. Optimal transport, entropy, and an overview of our method
  5. Dimensional convexity properties of relative entropy for log-concave measures with homogeneous potential
  6. Improvements in the Gauss space
  7. Applications to functional inequalities

Key Result

Theorem 1.1

Let $\nu$ be a Borel measure on $\mathbb{R}^{n}$ with density $\phi$ with respect to the Lebesgue measure, and fix $a\in[-\infty,\frac{1}{n}]$. Then the following are equivalent:

Theorems & Definitions (70)

  • Theorem 1.1
  • Conjecture 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5: An HWI inequality
  • Corollary 1.6: A logarithmic Sobolev inequality with an application
  • Corollary 1.7: A Talagrand inequality
  • Definition 2.1
  • Theorem 2.2
  • Example 2.3
  • ...and 60 more