In this paper, we focus on the following general shape optimization problem: min{J(Omega), Omega convex, Omega is an element of S(ad)}, where S(ad) is a set of 2-dimensional admissible shapes and J : Sad. R is a shape functional. Using a specific parameterization of the set of convex domains, we derive some extremality conditions (first and second order) for this kind of problem. Moreover, we use these optimality conditions to prove that, for a large class of functionals (satisfying a concavity-like property), any solution to this shape optimization problem is a polygon.