# Optimal convex shapes for concave functionals

Dorin Bucur; Ilaria Fragalà; Jimmy Lamboley

ESAIM: Control, Optimisation and Calculus of Variations (2012)

- Volume: 18, Issue: 3, page 693-711
- ISSN: 1292-8119

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topBucur, Dorin, Fragalà, Ilaria, and Lamboley, Jimmy. "Optimal convex shapes for concave functionals." ESAIM: Control, Optimisation and Calculus of Variations 18.3 (2012): 693-711. <http://eudml.org/doc/277816>.

@article{Bucur2012,

abstract = {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 Pólya-Szegö problem. },

author = {Bucur, Dorin, Fragalà, Ilaria, Lamboley, Jimmy},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Convex bodies; concavity inequalities; optimization; shape derivatives; capacity; convex bodies},

language = {eng},

month = {11},

number = {3},

pages = {693-711},

publisher = {EDP Sciences},

title = {Optimal convex shapes for concave functionals},

url = {http://eudml.org/doc/277816},

volume = {18},

year = {2012},

}

TY - JOUR

AU - Bucur, Dorin

AU - Fragalà, Ilaria

AU - Lamboley, Jimmy

TI - Optimal convex shapes for concave functionals

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2012/11//

PB - EDP Sciences

VL - 18

IS - 3

SP - 693

EP - 711

AB - 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 Pólya-Szegö problem.

LA - eng

KW - Convex bodies; concavity inequalities; optimization; shape derivatives; capacity; convex bodies

UR - http://eudml.org/doc/277816

ER -

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