A spectral isoperimetric inequality on the n-sphere for the Robin-Laplacian with negative boundary parameter

For every given <0, we study the problem of maximizing the first Robin eigenvalue of the Laplacian _ () among convex (not necessarily smooth) sets ^n with fixed perimeter. In particular, denoting by ₙ the perimeter of the n-dimensional hemisphere, we show that for fixed perimeters P<ₙ, geodesic balls maximize the eigenvalue. Moreover, we prove a quantitative stability result for this isoperimetric inequality in terms of volume difference between and the ball D of the same perimeter.