We focus here on the analysis of the regularity or singularity of solutions
$\Om_{0}$ to shape optimization problems among convex planar sets, namely: $$
J(\Om_{0})=\min\{J(\Om),\ \Om\ \textrm{convex},\ \Omega\in\mathcal S_{ad}\}, $$
where $\mathcal S_{ad}$ is a set of 2-dimensional admissible shapes and
$J:\mathcal{S}_{ad}\rightarrow\R$ is a shape functional. Our main goal is to
obtain qualitative properties of these optimal shapes by using first and second
order optimality conditions, including the infinite dimensional Lagrange
multiplier due to the convexity constraint.