Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Duality and Heat flow

Dario Cordero-Erausquin, Nathael Gozlan, Shohei Nakamura, Hiroshi Tsuji

TL;DR

This work connects Legendre duality for log-concave functions with Gaussian heat flow to derive a streamlined semigroup proof of the functional Blaschke-Santaló inequality. The core technique is an evolution law for the Legendre transform: if $f=e^{-}$ and $_t=-\\log P_t f$, then $\\u001f_t=(\\u001f_t)^*$ satisfies $\\partial_t \\u001f_t(z)=|z|^2-\\mathrm{Tr}(D^2 \\u001f_t(z))^{-1}$, with FP-flow analogues. They prove monotonicity of the volume product $M(f)=\\int f \\int f^\circ$ along the heat and FP flows for even $f$, with strict increase unless $f$ is a centered Gaussian, implying Gaussian local maximizers and the global maximizer in this setting. The paper also outlines a direct treatment of non-even cases via Laplace-transform type arguments, and emphasizes linear invariance as a structural feature that clarifies the semi-group approach.

Abstract

We reveal the relation between the Legendre transform of convex functions and heat flow evolution, and how it applies to the functional Blaschke-Santalo inequality. We also describe local maximizers in this inequality.

Duality and Heat flow

TL;DR

This work connects Legendre duality for log-concave functions with Gaussian heat flow to derive a streamlined semigroup proof of the functional Blaschke-Santaló inequality. The core technique is an evolution law for the Legendre transform: if and , then satisfies , with FP-flow analogues. They prove monotonicity of the volume product along the heat and FP flows for even , with strict increase unless is a centered Gaussian, implying Gaussian local maximizers and the global maximizer in this setting. The paper also outlines a direct treatment of non-even cases via Laplace-transform type arguments, and emphasizes linear invariance as a structural feature that clarifies the semi-group approach.

Abstract

We reveal the relation between the Legendre transform of convex functions and heat flow evolution, and how it applies to the functional Blaschke-Santalo inequality. We also describe local maximizers in this inequality.
Paper Structure (4 sections, 3 theorems, 40 equations)

This paper contains 4 sections, 3 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Heat flow and Legendre transform
  3. Blashke-Santaló inequality and Heat flow
  4. Remarks on the non-even case

Key Result

Proposition 2

Let $\phi:\mathbb{R}^n \to \mathbb{R}\cup\{+\infty\}$ be a super-linear convex function with $\int e^{-\phi}>0$. Consider the convex function $\phi_t = - \log P_t (e^{-\phi})$ where $P_t$ is the Heat semi-group and let $\psi_t = (\phi_t)^*$ be its Legendre transform. Then for every $z\in \mathbb{R}^

Theorems & Definitions (6)

  • Proposition 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof