On convex sets that minimize the average distance.

Antoine Lemenant (1) , Edoardo Mainini (2)

1 LJLL - Laboratoire Jacques-Louis Lions
2 Dipartimento di matematica F. Casorati

Abstract: In this paper we study the compact and convex sets K in the plane, that minimize the average of dist(x,K) with respect to x, and according to a given probability measure, and with volume or perimeter constraints on the set K. 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 we can prove that any arbitrary convex set can be a minimizer when both perimeter and volume constraints are considered.

Mots clés: Gamma-convergence; Shape optimization; Distance functional; Optimality conditions; Convex analysis; Second order variation; Gamma-convergence.

Domaines: Analyse numérique [cs.NA]

Antoine Lemenant, Edoardo Mainini. On convex sets that minimize the average distance.. 2011. ⟨hal-00564109⟩