New isoperimetric estimates for solutions to Monge-Ampere equations

We prove some sharp estimates for solutions to Dirichlet problems relative to Monge-Ampere equations. Among them we show that the eigenvalue of the Dirichlet problem, when computed on convex domains with fixed measure, is maximal on ellipsoids. This result falls in the class of affine isoperimetric inequalities and shows that the eigenvalue of the Monge-Ampere operator behaves just the contrary of the first eigenvalue of the Laplace operator. (C) 2008 Elsevier Masson SAS. All rights reserved.