  The classical Faber-Krahn inequality asserts that balls (uniquely) minimize
the first eigenvalue of the Dirichlet-Laplacian among sets with given volume.
In this paper we prove a sharp quantitative enhancement of this result, thus
confirming a conjecture by Nadirashvili and Bhattacharya-Weitsman. More
generally, the result applies to every optimal Poincar\'e-Sobolev constant for
the embeddings $W^{1,2}_0(\Omega)\hookrightarrow L^q(\Omega)$.
