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

### Xin Yang Lu

Carnegie Mellon University, Pittsburgh, USA

## 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, $d\geq 2$, $\mu$ 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.

## Cite this article

Xin Yang Lu, Example of minimizer of the average-distance problem with non closed set of corners. Rend. Sem. Mat. Univ. Padova 137 (2017), pp. 19–55

DOI 10.4171/RSMUP/137-2