Shape optimization under convexity constraint

The shape optimization is the study of optimization problems whose unknown is a domain of $ R^d$. I will focus on the case where admissibles shapes are required to be convex sets of $ R2$. Under this constraint, it is hard to write optimality conditions. In a first part, I will show how we can write such conditions (first and second order), and I will use these ones to exhibit a class of functionals which leads to polygonal optimal shapes (work with A. Novruzi). In a second part, I will focus on the minimization of the second eigenvalue for the Laplace operator (Dirichlet conditions), model problem which shows difficulties linked to convexity constraint, and also difficulties due to the regularity of optimal shapes. We particularly show that optimal shapes are C^{1,1/2} and no more, for this problem. I end with some links with partially overdetermined problems.