Shape Optimization Problems over Classes of Convex Domains

We consider shape optimization problems of the form $\min\{\int_{\partial A} f(x,\nu(x))\,\mathrm{d}\mathcal{H}^{n-1} : A \in \mathcal{A}\}$, where f is any continuous function and the class 𝒜 of admissible domains is made of convex sets. We prove the existence of an optimal solution provided the domains satisfy some suitable constraints.