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Publication 14-CNA-002

Example of Minimizer of the Average Distance Problem with Non Closed Set of Corners

Xin Yang Lu
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
xinyang@andrew.cmu.edu

Abstract: The average distance problem, in the penalized formulation, involves minimizing $$ {\rm min} \ E_{\mu}^{\lambda},\qquad E_{\mu}^{\lambda}(\sum):=\int d(x,\sum)d\mu(x)+\lambda {{\cal H}^1(\sum)}, $$ among path-wise connected, closed sets $\sum$ with finite ${\cal H}^1$-measure, where $d(x,\sum):=d_{\cal H}(\{x\},\sum)$ (with $d_{\cal H}$ denoting the Hausdorff distance), and $\mu$ is a given probability measure. Regularity of minimizers has been a rather difficult problem. It is known that minimizers can fail to be $C^1$ regular, i.e. there exist minimizers containing corners. An interesting question is whether the set of corners is closed. The aim of this paper is to provide an example of minimizer whose set of corners is countable and not closed.

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