More on functional and quantitative versions of the isoperimetric inequality

TL;DR

This paper introduces new functional and quantitative forms of the isoperimetric inequality using optimal transportation, providing refined bounds and applications in convex geometry.

Contribution

It presents novel functional and quantitative versions of the isoperimetric inequality utilizing optimal transportation techniques.

Findings

New functional form of the isoperimetric inequality

Quantitative inequality with Wasserstein distance

Applications in convex geometry

Abstract

This paper deals with the famous isoperimetric inequality. In a first part, we give some new functional form of the isoperimetric inequality, and in a second part, we give a quantitative form with a remainder term involving Wasserstein distance of the classical isopemetric inequality. In both parts, we use optimal transportation. Finally, we use our refined isoperimetric inequality in some classical cases arising in convex geometry.