Regularity of Optimal Convex Shapes

We consider shape optimisation problems in the class of convex sets. Assuming that the shape functional satisfies a Lipschitz like property with respect to a distance issued from the γ-convergence, we prove that the minimiser has the boundary of class C1. In particular, we prove that large classes of functionals depending on the eigenvalues of the Dirichlet Laplacian satisfy this property. The key point of the paper is the understanding of the asymptotic behaviour of the γ-convergence near the "angular" points of the convex set.