The Brunn-Minkowski inequality for the Monge-Ampère eigenvalue and smoothness of the eigenfunctions

Abstract

The original form of the Brunn-Minkowski inequality involves volumes of convex bodies in ${ℝ}^n$ and states that the $n$-th root of the volume is a concave function with respect to the Minkowski addition of convex bodies.

In 1976, Brascamp and Lieb proved a Brunn-Minkowski inequality for the first eigenvalue of the Laplacian. In this talk, I will discuss a nonlinear analogue of the above result, that is, the Brunn-Minkowski inequality for the eigenvalue of the Monge-Ampère operator. For this purpose, I will first introduce the Monge-Ampère eigenvalue problem on general bounded convex domains. Then, I will present several properties of the eigenvalues and related analysis concerning smoothness of the eigenfunctions.