A note on some Poincar\'e inequalities on convex sets by Optimal Transport methods

TL;DR

This paper demonstrates how optimal transport methods can derive Poincaré-Wirtinger inequalities on convex sets, including unbounded ones, expanding the theoretical toolkit for analysis on such domains.

Contribution

It introduces a novel approach using dynamical optimal transport to establish Poincaré inequalities on convex sets, generalizing previous results to unbounded domains.

Findings

Poincaré-Wirtinger inequalities derived via optimal transport

Applicable to both bounded and unbounded convex sets

Provides a new proof technique for functional inequalities

Abstract

We show that a class of Poincar\'e-Wirtinger inequalities on bounded convex sets can be obtained by means of the dynamical formulation of Optimal Transport. This is a consequence of a more general result valid for convex sets, possibly unbounded.