Mathematics

We review various geometric inequalities, and discuss the rigidity and stability problems associated with them. A list of tentative topics would be: The Euclidean isoperimetric inequality, proof by Steiner's symmetrization. Sharp quantitative version by symmetrization, by mass transportation and by selection principle. Almgren's isoperimetric inequality and the Michael-Simon conjecture. Alexandrov's theorem, proof by moving planes, by tubular neighborhoods, by the Heintze-Karcher inequality. Stability for Alexandrov's theorem. The Sobolev inequality, Obata's theorem and stability/rigidity issues for constant scalar curvature.