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

Lu, Xin Yang

Rendiconti del Seminario Matematico della Università di Padova, Tome 137 (2017), pp. 19-55.

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

$$E_{\mu }^{\lambda }\left(\Sigma \right):={\int }_{{ℝ}^d}{\inf }_{y\in \Sigma }|x-y|\mathrm{d}\mu \left(x\right)+\lambda {\mathscr{H}}^1\left(\Sigma \right),$$

among compact, connected sets $\Sigma$, where ${\mathscr{H}}^1$ 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.