On convex sets that minimize the average distance

Abstract

In this paper we study the compact and convex sets K R 2 that minimize Z dist(x;K)dx + 1V ol(K) + 2Per(K) for some constants 1 and 2, that could eventually 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 dierently since polygons are never minimizers. Finally using a purely geometrical argument from Tilli [21] we can prove that any arbitrary convex set can be a minimizer when both perimeter and volume constraints are considered.