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Optimal convex shapes for concave functionals

Dorin BucurIlaria FragalàJimmy Lamboley — 2012

ESAIM: Control, Optimisation and Calculus of Variations

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...

Optimal convex shapes for concave functionals

Dorin BucurIlaria FragalàJimmy Lamboley — 2012

ESAIM: Control, Optimisation and Calculus of Variations

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...

Optimal convex shapes for concave functionals

Dorin BucurIlaria FragalàJimmy Lamboley — 2012

ESAIM: Control, Optimisation and Calculus of Variations

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...

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