Optimal convex shapes for concave functionals

TL;DR

This paper investigates the concavity properties of shape functionals, introduces a new algebraic structure for convex bodies, and applies second order shape derivatives to analyze optimal shapes, especially in relation to the Polya-Szeg"o conjecture.

Contribution

It introduces a novel algebraic framework for convex bodies, providing new concavity and indecomposability results, and applies second order shape derivatives to analyze optimal convex shapes.

Findings

Counterexamples to Blaschke-concavity of capacity.

A new algebraic structure enabling global concavity results.

Exclusion of smooth, positively curved regions in optimal shapes.

Abstract

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.