Calculus of Variations and Geometric Measure Theory
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A. Lemenant - E. Mainini

On convex sets that minimize the average distance

created by mainini on 07 Sep 2010
modified on 26 Mar 2011


Accepted Paper

Inserted: 7 sep 2010
Last Updated: 26 mar 2011

Journal: ESAIM: COCV
Year: 2011


In this paper we study the convex sets that minimize the average distance functional in dimension 2, with volume or perimeter penalization. We compute in particular the second order derivative of the functional and use it to exclude positive curvature points for the problem with volume constraint. The problem with perimeter constraint behaves differently since polygons are never minimizers. Finally using a purely geometrical argument of Paolo Tilli we can prove that any arbitrary convex set can be a minimizer when both perimeter and volume constraints are considered.


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