Some weighted isoperimetric inequalities in quantitative form

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.

We spotted a mistake in the proof of lemma 3.1