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 $$ E\\mu^\\lambda(\\Sigma):=\\int{\\mathbb R^d} \\mathrm {inf}\_{y\\in\\Sigma} |x-y|\\mathrm d\\mu(x)+\\lambda\\mathcal H^1(\\Sigma), $$ among compact, connected sets $\\Sigma$, where $\\mathcal H^1$ denotes the 1-Hausdorff measure, $d\\geq 2$, $\\mu$ 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.