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

A sharp estimate for the first Robin-Laplacian eigenvalue with negative boundary parameter