Volume 18, Number 4, October-December 2012
|Page(s)||1049 - 1072|
|Published online||16 January 2012|
On convex sets that minimize the average distance
UniversitéParis Diderot – Paris 7, U.F.R de
Mathématiques, Site Chevaleret,
Case 7012, 75205
Paris Cedex 13,
2 Dipartimento di Matematica ‘F. Casorati’, Universià degli Studi di Pavia, via Ferrata 1, 27100 Pavia, Italy
In this paper we study the compact and convex sets K ⊆ Ω ⊆ ℝ2 that minimize for some constants λ1 and λ2, that could possibly be zero. We compute in particular the second order derivative of the functional and use it to exclude smooth points of positive curvature for the problem with volume constraint. The problem with perimeter constraint behaves differently since polygons are never minimizers. Finally using a purely geometrical argument from Tilli [J. Convex Anal. 17 (2010) 583–595] we can prove that any arbitrary convex set can be a minimizer when both perimeter and volume constraints are considered.
Mathematics Subject Classification: 49Q10 / 49K30
Key words: Shape optimization / distance functional / optimality conditions / convex analysis / second order variation / gamma-convergence
© EDP Sciences, SMAI, 2012
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