  Motivated by a long-standing conjecture of Polya and Szeg\"o 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 isoperimetriclike inequalities. As a byproduct of this approach
we also obtain a quantitative version of the Kneser-S\"uss 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 Polya-Szeg\"o problem.
