  In this paper we prove that the ball maximizes the first eigenvalue of the
Robin Laplacian operator with negative boundary parameter, among all convex
sets of \mathbb{R}^n with prescribed perimeter. The key of the proof is a
dearrangement procedure of the first eigenfunction of the ball on the level
sets of the distance function to the boundary of the convex set, which controls
the boundary and the volume energies of the Rayleigh quotient.

Analysis of PDEs;Optimization and Control