The Stability of the Isoperimetric Inequality

These notes contain five lectures given by the author at the CNA Summer School held at Carnegie Mellon University in Pittsburgh from May 30 to June 7, 2013. The aim of the course was to give a self contained introduction to the classical isoperimetric inequality and to various stability results proved in recent years for this inequality and other related geometric and analytic inequalities. In the first lecture we recall the basic definitions and the main properties of sets of finite perimeter: the structure theorem, approximation with smooth sets, compactness. These are the main ingredients of De Giorgi’s proof of the isoperimetric inequality via Steiner symmetrization. This result is established in the second lecture which also contains the coarea formula for both functions and sets of finite perimeter and the proof of the equivalence between isoperimetric and Sobolev inequality. With the third lecture we enter into the subject of the stability. There, we give the complete proof of the quantitative estimates obtained by Fuglede for convex and nearly spherical sets. In Lecture 4 we give a detailed account of the proof of the quantitative isoperimetric inequality for general sets first obtained by F., Maggi and Pratelli. This proof is based on symmetrization arguments that also apply to a variety of other inequalities. In the last lecture two different proofs are presented. The first one, due to Figalli, Maggi and Pratelli starts from Gromov’s proof of the isoperimetric property of the balls and uses Brenier optimal transportation map between sets. The second one, due to Cicalese and Leonardi, is based on the regularity theory for area almost minimizers. Most of the notes were taken by Ryan Murray who typed them live in the classroom. Matteo Rinaldi added some extra material from some hand written notes of mine and did the editing. Laura Bufford did all the pictures. I warmly thank them for the excellent job. Without their precious help probably these notes would have never appeared.