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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...
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...
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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