Quantitative Isoperimetric Inequalities on the Real Line

TL;DR

This paper extends sharp quantitative isoperimetric inequalities from Gauss space to all symmetric log-concave measures on the real line, identifying intervals or their complements as minimal perimeter sets for given measure and asymmetry.

Contribution

It provides a geometric proof of isoperimetric inequalities for symmetric log-concave measures, generalizing previous results from Gauss space.

Findings

Intervals or complements minimize perimeter for given measure and asymmetry.

Established sharp quantitative inequalities for symmetric log-concave measures.

Extended prior Gauss space results to a broader class of measures.

Abstract

In a recent paper A. Cianchi, N. Fusco, F. Maggi, and A. Pratelli have shown that, in the Gauss space, a set of given measure and almost minimal Gauss boundary measure is necessarily close to be a half-space. Using only geometric tools, we extend their result to all symmetric log-concave measures \mu on the real line. We give sharp quantitative isoperimetric inequalities and prove that among sets of given measure and given asymmetry (distance to half line, i.e. distance to sets of minimal perimeter), the intervals or complements of intervals have minimal perimeter.