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.

This paper considers Schrödinger operators, and presents a probabilistic interpretation of the variation (or shape derivative) of the Dirichlet groundstate energy when the associated domain is perturbed. This interpretation relies on the distribution on the boundary of a stopped random process with Feynman-Kac weights. Practical computations require in addition the explicit approximation of the normal derivative of the groundstate on the boundary. We then propose to use this formulation in the case of the so-called fixed node approximation of Fermion groundstates, defined by the bottom eigenelements of the Schrödinger operator of a Fermionic system with Dirichlet conditions on the nodes (the set of zeros) of an initially guessed skew-symmetric function. We show that shape derivatives of the fixed node energy vanishes if and only if either (i) the distribution on the nodes of the stopped random process is symmetric; or (ii) the nodes are exactly the zeros of a skew-symmetric eigenfunction of the operator. We propose an approximation of the shape derivative of the fixed node energy that can be computed with a Monte-Carlo algorithm, which can be referred to as Nodal Monte-Carlo (NMC). The latter approximation of the shape derivative also vanishes if and only if either (i) or (ii) holds.