Stability of isoperimetric type inequalities for some Monge–Ampère functionals

Abstract

We investigate the stability of some inequalities of isoperimetric type related to Monge–Ampere functionals. In particular, firstly we prove the stability of a reverse Faber–Krahn inequality for the Monge–Ampere eigenvalue and its generalization. Then we give a stability result for the Brunn–Minkowski inequality and for a consequent Urysohn’s type inequality for the so-called \(n\)-torsional rigidity, a natural extension of the usual torsional rigidity.