From Brunn-Minkowski to Pr\'ekopa-Leindler and Borell-Brascamp-Lieb: discrete inequalities

TL;DR

This paper develops a unified approach to derive Prékopa-Leindler and Borell-Brascamp-Lieb inequalities from Brunn-Minkowski inequalities, extending these concepts to discrete settings over integer lattices with new proofs and examples.

Contribution

It introduces a general method to obtain classical inequalities from Brunn-Minkowski inequalities and proves their discrete analogues for functions over Z^d, advancing the understanding of discrete convex geometry.

Findings

Derived discrete Prékopa-Leindler and Borell-Brascamp-Lieb inequalities.

Provided numerous examples illustrating the inequalities.

Extended classical inequalities to the discrete setting over Z^d.

Abstract

We consider a general way to obtain Pr\'ekopa-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 Pr\'ekopa-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.