Regularity in shape optimization under convexity constraint

This paper is concerned with the regularity of shape optimizers of a class of isoperimetric problems under convexity constraint. We prove that minimizers of the sum of the perimeter and a perturbative term, among convex shapes, are C 1 , 1 -regular. To that end, we define a notion of quasi-minimizer fitted to the convexity context and show that any such quasi-minimizer is C 1 , 1 -regular. The proof relies on a cutting procedure which was introduced to prove similar regularity results in the calculus of variations context. Using a penalization method we are able to treat a volume constraint, showing the same regularity in this case. We go through some examples taken from PDE theory, that is when the perturbative term is of PDE type, and prove that a large class of such examples fit into our C 1 , 1 -regularity result. Finally we provide a counter-example showing that we cannot expect higher regularity in general.