Bucur, Dorin; Fragalà, Ilaria; Lamboley, Jimmy Optimal convex shapes for concave functionals. (English) Zbl 1253.49031 ESAIM, Control Optim. Calc. Var. 18, No. 3, 693-711 (2012). Summary: 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 the Pólya-Szegö problem. Cited in 1 ReviewCited in 9 Documents MSC: 49Q10 Optimization of shapes other than minimal surfaces 31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions Keywords:convex bodies; concavity inequalities; optimization; shape derivatives; capacity PDFBibTeX XMLCite \textit{D. Bucur} et al., ESAIM, Control Optim. Calc. Var. 18, No. 3, 693--711 (2012; Zbl 1253.49031) Full Text: DOI arXiv