Quantitative isoperimetric inequalities for log-convex probability measures on the line

TL;DR

This paper investigates the isoperimetric inequality for symmetric log-convex probability measures on the line, revealing the structure of extremal sets and a surprising discrepancy between near-optimality and set proximity.

Contribution

It provides a new quantitative form of the isoperimetric inequality for log-convex measures, highlighting anomalous behaviors and deriving related functional inequalities.

Findings

Extremal sets are intervals or their complements.

Near-optimal sets can be far from extremal sets in symmetric difference.

The results lead to new weak Cheeger type inequalities.

Abstract

The purpose of this paper is to analyze the isoperimetric inequality for symmetric log-convex probability measures on the line. Using geometric arguments we first re-prove that extremal sets in the isoperimetric inequality are intervals or complement of intervals (a result due to Bobkov and Houdr\'e). Then we give a quantitative form of the isoperimetric inequality, leading to a somehow anomalous behavior. Indeed, it could be that a set is very close to be optimal, in the sense that the isoperimetric inequality is almost an equality, but at the same time is very far (in the sense of the symmetric difference between sets) to any extremal sets! From the results on sets we derive quantitative functional inequalities of weak Cheeger type.