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

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