Motivated by a long-standing conjecture of Pólya and Szegö about the Newtonian capacityof convex bodies, we discuss the role of concavity inequalities in shape optimization, andwe provide several counterexamples to the Blaschke-concavity of variational functionals,including capacity. We then introduce a new algebraic structure on convex bodies, whichallows to obtain global concavity and indecomposability results, and we discuss theirapplication to isoperimetric-like inequalities. As a byproduct of this approach we alsoobtain a quantitative version of the Kneser-Süss inequality. Finally, for a large class offunctionals involving Dirichlet energies and the surface measure, we perform a localanalysis of strictly convex portions of the boundary via second ordershape derivatives. This allows in particular to exclude the presence of smooth regionswith positive Gauss curvature in an optimal shape for Pólya-Szegö problem.
