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Functional Liftings of Restricted Geometric Inequalities

Andreas Malliaris, James Melbourne, Cyril Roberto, Michael Roysdon

TL;DR

This work builds a unifying framework that connects geometric inequalities and their functional counterparts through generalized sup-convolutions, providing a robust method to translate between set-based and function-based Brunn–Minkowski type statements. It proves an abstract equivalence theorem showing that geometric inequalities for sets imply corresponding functional inequalities, and vice versa, in a broad class of measure spaces and groups. The authors establish a Gaussian dimensional Brunn–Minkowski inequality for unimodal functions, a functional version of the log-Brunn–Minkowski inequality, and functional $L_p$-Brunn–Minkowski inequalities for both $p\in[0,1)$ and $p>1$, including a Borell–Brascamp–Lieb type inequality on nilpotent Lie groups. The results extend the equivalence between geometric and functional inequalities to abstract and non-Euclidean settings, enabling transfer of inequalities across geometry, analysis, and group theory.

Abstract

We investigate what we term "generalized sup-convolutions". We show that functional inequalities that enjoy an interpretation as sup-convolution inequalities can be deduced from the special case of indicator functions corresponding to a geometric inequality. As consequences we derive a Borell-Brascamp Lieb inequality for the Gaussian Brunn-Minkowski inequality and give a functional analog of the log-Brunn Minkowski conjecture. Though we focus on Euclidean applications, our results are general and can be directly applied in more abstract settings, like groups or even topological measure spaces without algebraic structure, we instantiate this claim with a Borell-Brascamp-Lieb type inequality for nilpotent Lie groups.

Functional Liftings of Restricted Geometric Inequalities

TL;DR

This work builds a unifying framework that connects geometric inequalities and their functional counterparts through generalized sup-convolutions, providing a robust method to translate between set-based and function-based Brunn–Minkowski type statements. It proves an abstract equivalence theorem showing that geometric inequalities for sets imply corresponding functional inequalities, and vice versa, in a broad class of measure spaces and groups. The authors establish a Gaussian dimensional Brunn–Minkowski inequality for unimodal functions, a functional version of the log-Brunn–Minkowski inequality, and functional -Brunn–Minkowski inequalities for both and , including a Borell–Brascamp–Lieb type inequality on nilpotent Lie groups. The results extend the equivalence between geometric and functional inequalities to abstract and non-Euclidean settings, enabling transfer of inequalities across geometry, analysis, and group theory.

Abstract

We investigate what we term "generalized sup-convolutions". We show that functional inequalities that enjoy an interpretation as sup-convolution inequalities can be deduced from the special case of indicator functions corresponding to a geometric inequality. As consequences we derive a Borell-Brascamp Lieb inequality for the Gaussian Brunn-Minkowski inequality and give a functional analog of the log-Brunn Minkowski conjecture. Though we focus on Euclidean applications, our results are general and can be directly applied in more abstract settings, like groups or even topological measure spaces without algebraic structure, we instantiate this claim with a Borell-Brascamp-Lieb type inequality for nilpotent Lie groups.

Paper Structure

This paper contains 7 sections, 23 theorems, 141 equations.

Table of Contents

  1. Introduction
  2. Generalized Sup-Convolutions
  3. Dimensional Functional Brunn-Minkowksi inequalities in measure spaces
  4. Functional $L_p$-Brunn-Minkowski inequalities
  5. The case $p \in [0,1)$
  6. The case $p > 1$
  7. An Abstract Approach to the Equivalence of Geometric and Functional Inequalities

Key Result

Theorem 1.1

Let $\mu$ be the measure on $\mathbb{R}^n$, with density $e^{-w(|x|)}$, where $w:[0,\infty)\to (-\infty,\infty]$ is increasing and $\mathbb{R}\ni t \mapsto w(e^t)$ is convex. Let $t\in (0,1)$, $\alpha\geq -\frac{1}{n}$. If $f,g:\mathbb{R}^n\to [0,\infty)$ are even unimodal and $h:\mathbb{R}^n\to [0, then where $\beta=\frac{\alpha}{1+n\alpha}$. Moreover, eq:Theorem1.1 is equivalent to the geometri

Theorems & Definitions (48)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • proof : Proof of Theorem \ref{['thm: everything is sup convolution']}
  • Theorem 2.4
  • Lemma 2.5
  • ...and 38 more