On convex sets that minimize the average distance

Lemenant, Antoine; Mainini, Edoardo

ESAIM: Control, Optimisation and Calculus of Variations, Tome 18 (2012) no. 4, pp. 1049-1072.

In this paper we study the compact and convex sets $K\subset \Omega \subset {ℝ}^2$ that minimize

$${\int }_{\Omega }\mathrm{dist}(𝐱,K)d𝐱+{\lambda }_1\mathrm{Vol}\left(K\right)+{\lambda }_2\mathrm{Per}\left(K\right)$$

for some constants ${\lambda }_1$ and ${\lambda }_2$, 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.

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