Example of minimizer of the average-distance problem with non closed set of corners

Abstract

The average-distance problem, in the penalized formulation, involves minimizing

among compact, connected sets $\Sigma$, where ${\mathcal{H}}^1$ denotes the 1-Hausdorff measure, $d\ge 2$, $\mu$ is a given measure and $\lambda$ a given parameter. Regularity of minimizers is a delicate problem. It is known that even if $\mu$ is absolutely continuous with respect to Lebesgue measure, $C^1$ regularity does not hold in general. An interesting question is whether the set of corners, i.e. points where $C^1$ regularity does not hold, is closed. The aim of this paper is to provide an example of minimizer whose set of corners is not closed, with reference measure $\mu$ absolutely continuous with respect to Lebesgue measure.