On convex sets that minimize the average distance

¹ UniversitéParis Diderot – Paris 7, U.F.R de Mathématiques, Site Chevaleret, Case 7012, 75205 Paris Cedex 13, France
lemenant@ann.jussieu.fr
² Dipartimento di Matematica ‘F. Casorati’, Universià degli Studi di Pavia, via Ferrata 1, 27100 Pavia, Italy
edoardo.mainini@unipv.it

Received: 7 September 2010

Abstract

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.