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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 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. Get the paper in its entirety as |