Some weighted isoperimetric inequalities in quantitative form

TL;DR

This paper investigates weighted isoperimetric inequalities, establishing sharp stability results for log-convex weights and analyzing the effects of negative power weights to comprehensively understand the problem.

Contribution

It provides new stability results for log-convex weights and a complete analysis of isoperimetric inequalities with negative power weights.

Findings

Sharp stability result for log-convex weight inequality

Behavior analysis of negative power weight in perimeter

Comprehensive understanding of weighted isoperimetric problems

Abstract

In this paper we study two different weighted isoperimetric inequalities. In the first part of the paper we prove a sharp stability result for the isoperimetric inequality with a log-convex weight. In the second part we analize the behavior of a negative power weight for the perimeter thus providing a complete picture of the isoperimetric problem in this context.