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On convex sets that minimize the averagedistance

Published online by Cambridge University Press:  16 January 2012

Antoine Lemenant  and
Edoardo Mainini
Antoine Lemenant
Affiliation:
UniversitéParis Diderot – Paris 7, U.F.R de Mathématiques, Site Chevaleret, Case 7012, 75205 Paris Cedex 13, France. [email protected]
Edoardo Mainini
Affiliation:
Dipartimento di Matematica ‘F. Casorati’, Universià degli Studi di Pavia, via Ferrata 1, 27100 Pavia, Italy; [email protected]
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Abstract

In this paper we study the compact and convex setsK ⊆ Ω ⊆ ℝ2 that minimize

\begin{equation*} \int_{\Omega} \dist(\x,K) \,{\rm d}\x +\lambda_1 {\rm Vol}(K)+\lambda_2 {\rm Per}(K) \end{equation*}∫Ωdist(x,K)dx+λ1Vol(K)+λ2Per(K)
for some constants λ1 andλ2, that could possibly be zero. We compute in particularthe second order derivative of the functional and use it to exclude smooth points ofpositive curvature for the problem with volume constraint. The problem with perimeterconstraint behaves differently since polygons are never minimizers. Finally using a purelygeometrical argument from Tilli [J. Convex Anal. 17 (2010)583–595] we can prove that any arbitrary convex set can be a minimizer when both perimeterand volume constraints are considered.

Keywords

Shape optimizationdistance functionaloptimality conditionsconvex analysissecond order variationgamma-convergence
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 18 , Issue 4 , October 2012 , pp. 1049 - 1072
Copyright
© EDP Sciences, SMAI, 2012

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