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Remarks on Brunn-Minkowski-type inequalities related to the Ornstein-Uhlenbeck operator

Francisco Marín Sola, Francesco Salerno

Abstract

We investigate Brunn-Minkowski-type inequalities for the torsional rigidity $T_γ$ and the first eigenvalue $λ_γ$ associated with the Ornstein-Uhlenbeck operator. Counterexamples are provided showing that neither concavity nor convexity properties hold for $T_γ$ on general bounded convex sets. We also demonstrate that log-concavity and log-convexity properties fail in this setting. In the case of centrally symmetric sets, we answer a question raised by Cordero-Erausquin and Eskenazis by showing that $T_γ^{1/(n+2)}$ is neither convex nor concave. On the positive side, we prove that $T_γ^{1/3}$ is convex with respect to Minkowski addition when restricted to Euclidean balls centered at the origin. For $λ_γ$, we answer negatively a question posed by Colesanti, Francini, Livshyts, and Salani by showing that the inequality $λ_γ(Ω_t)^{-1/2} \geq (1-t)λ_γ(Ω_0)^{-1/2} + tλ_γ(Ω_1)^{-1/2}$ does not hold, even for centrally symmetric sets.

Remarks on Brunn-Minkowski-type inequalities related to the Ornstein-Uhlenbeck operator

Abstract

We investigate Brunn-Minkowski-type inequalities for the torsional rigidity and the first eigenvalue associated with the Ornstein-Uhlenbeck operator. Counterexamples are provided showing that neither concavity nor convexity properties hold for on general bounded convex sets. We also demonstrate that log-concavity and log-convexity properties fail in this setting. In the case of centrally symmetric sets, we answer a question raised by Cordero-Erausquin and Eskenazis by showing that is neither convex nor concave. On the positive side, we prove that is convex with respect to Minkowski addition when restricted to Euclidean balls centered at the origin. For , we answer negatively a question posed by Colesanti, Francini, Livshyts, and Salani by showing that the inequality does not hold, even for centrally symmetric sets.
Paper Structure (5 sections, 11 theorems, 122 equations, 2 figures)

This paper contains 5 sections, 11 theorems, 122 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Counterexamples
  4. Main result
  5. Appendix

Key Result

Theorem 1.1

Let $\Omega_0$, $\Omega_1$ be open, bounded subsets of $\mathbb{R}^n$. Let $t\in[0,1]$. Then where

Figures (2)

  • Figure 1: Plot of $f(1,R)$ and $f(2,R)$.
  • Figure 2: Plot of $f(3,R)$, $f(5,R)$, and $f(10,R)$.

Theorems & Definitions (25)

  • Theorem 1.1: Theorem 1.2, CFLS
  • Theorem 1.2
  • Corollary 1.3
  • Conjecture 1.4
  • Proposition 1.5
  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.1
  • Example 3.1
  • Example 3.2
  • ...and 15 more