Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

From Brunn-Minkowski to Prékopa-Leindler and Borell-Brascamp-Lieb: discrete inequalities

Peter van Hintum

TL;DR

The paper develops discrete analogues of the Prékopa–Leindler and Borell–Brascamp–Lieb inequalities by deriving them from a discrete Brunn–Minkowski framework. It introduces a general lift principle that converts BM-type statements into BB/PL-type integral inequalities via level-sets and transport arguments, and shows that the discrete PL/BBL inequality arises as the $p o0$ limit of the BB L inequality under a non-degeneracy condition. A broad array of applications is then derived, including Gaussian, unconditional, compact Lie group, Riemannian manifold, and discrete cube settings, as well as a discrete analogue of Kemperman’s product-set theorem. The results unify discrete convex-geometric techniques with functional inequalities on $Z^d$ and related structures, expanding additive combinatorics and analysis on discrete spaces. Overall, the work provides a versatile, systematic method to transfer BM-type estimates into BB/PL-type inequalities in discrete environments, with sharp asymptotics and wide applicability.

Abstract

We consider a general way to obtain Prékopa-Leindler and Borell-Brascamp-Lieb type inequalities from Brunn-Minkowski type inequalities and provide numerous examples. We use the same heuristic to prove a discrete version of the Prékopa-Leindler and Borell-Brascamp-Lieb inequalities for functions over $\mathbb{Z}^d$. These are the functional extensions of the discrete Brunn-Minkowski inequality conjectured by Ruzsa and recently established by Keevash, Tiba, and the author.

From Brunn-Minkowski to Prékopa-Leindler and Borell-Brascamp-Lieb: discrete inequalities

TL;DR

The paper develops discrete analogues of the Prékopa–Leindler and Borell–Brascamp–Lieb inequalities by deriving them from a discrete Brunn–Minkowski framework. It introduces a general lift principle that converts BM-type statements into BB/PL-type integral inequalities via level-sets and transport arguments, and shows that the discrete PL/BBL inequality arises as the limit of the BB L inequality under a non-degeneracy condition. A broad array of applications is then derived, including Gaussian, unconditional, compact Lie group, Riemannian manifold, and discrete cube settings, as well as a discrete analogue of Kemperman’s product-set theorem. The results unify discrete convex-geometric techniques with functional inequalities on and related structures, expanding additive combinatorics and analysis on discrete spaces. Overall, the work provides a versatile, systematic method to transfer BM-type estimates into BB/PL-type inequalities in discrete environments, with sharp asymptotics and wide applicability.

Abstract

We consider a general way to obtain Prékopa-Leindler and Borell-Brascamp-Lieb type inequalities from Brunn-Minkowski type inequalities and provide numerous examples. We use the same heuristic to prove a discrete version of the Prékopa-Leindler and Borell-Brascamp-Lieb inequalities for functions over . These are the functional extensions of the discrete Brunn-Minkowski inequality conjectured by Ruzsa and recently established by Keevash, Tiba, and the author.

Paper Structure

This paper contains 12 sections, 20 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Inequalities in Euclidean space
  3. From Brunn-Minkowski to Borell-Brascamp-Lieb
  4. Examples
  5. Gaussian measure
  6. Dimensional Brunn-Minkowski conjecture
  7. Unconditional measures
  8. Kemperman's theorem
  9. Compact Lie groups
  10. Riemannian Manifolds
  11. Discrete Cube
  12. Proof of the discrete Borell-Brascamp-Lieb inequality

Key Result

Theorem 1.1

For all $d,\epsilon>0$, there exists a $n=n_{d,\epsilon}$ so that if $A,B\subset\mathbb{Z}^d$ are so that $B$ is not covered by $n$ parallel hyperplanes, then

Theorems & Definitions (29)

  • Theorem 1.1: van2025ruzsa
  • Theorem 1.2
  • Corollary 1.3
  • Proposition 1.4
  • proof
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Theorem 1.8: eskenazis2021dimensional
  • Corollary 1.9
  • ...and 19 more