# Transport Proofs Of Some Discrete Variants Of The Pr{\'e}Kopa-leindler Inequality

## TL;DR

This paper provides a transport-based proof of discrete displacement convexity of entropy on integers, leading to new discrete forms of the Prékopa-Leindler inequality, including the Four Functions Theorem and recent results on Z.

## Contribution

It introduces a novel transport proof for discrete displacement convexity, deriving new discrete inequalities from continuous analogs.

## Key findings

- Established a transport proof for discrete displacement convexity on integers
- Derived two discrete forms of the Prékopa-Leindler inequality
- Connected continuous and discrete convexity results

## Abstract

We give a transport proof of a discrete version of the displacement convexity of entropy on integers (Z), and get, as a consequence, two discrete forms of the Pr{\'e}kopa-Leindler Inequality : the Four Functions Theorem of Ahlswede and Daykin on the discrete hypercube [1] and a recent result on Z due to Klartag and Lehec [16].