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