# The eigenvalue problem for the Monge-Amp\`ere operator on general bounded convex domains

## TL;DR

This paper investigates the eigenvalue problem for the Monge-Ampère operator on general convex domains, establishing fundamental properties like existence, uniqueness, and stability, and extending previous smooth domain results to broader settings.

## Contribution

It extends the theory of Monge-Ampère eigenvalues to non-smooth convex domains, proving existence, uniqueness, and stability results that were previously known only for smooth, uniformly convex domains.

## Key findings

- Existence and uniqueness of Monge-Ampère eigenvalues and eigenfunctions on general convex domains.
- Stability of eigenvalues under Hausdorff convergence of domains.
- Application of results to geometric inequalities like Brunn-Minkowski and isoperimetric inequalities.

## Abstract

In this paper, we study the eigenvalue problem for the Monge-Amp\`ere operator on general bounded convex domains. We prove the existence, uniqueness and variational characterization of the Monge-Amp\`ere eigenvalue. The convex Monge-Amp\`ere eigenfunctions are shown to be unique up to positive multiplicative constants. Our results are the singular counterpart of previous results by P-L. Lions and K. Tso in the smooth, uniformly convex setting. Moreover, we prove the stability of the Monge-Amp\`ere eigenvalue with respect to the Hausdorff convergence of the domains. This stability property makes it possible to investigate the Brunn-Minkowski, isoperimetric and reverse isoperimetric inequalities for the Monge-Amp\`ere eigenvalue in their full generality. We also discuss related existence and regularity results for a class of degenerate Monge-Amp\`ere equations.