A quantitative reverse Faber-Krahn inequality for the first Robin eigenvalue with negative boundary parameter

The aim of this paper is to prove a quantitative form of a reverse Faber-Krahn type inequality for the first Robin Laplacian eigenvalue λβ with negative boundary parameter among convex sets of prescribed perimeter. In that framework, the ball is the only maximizer for λβ and the distance from the optimal set is considered in terms of Hausdorff distance. The key point of our stategy is to prove a quantitative reverse Faber-Krahn inequality for the first eigenvalue of a Steklov-type problem related to the original Robin problem.