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 , where denotes the 1-Hausdorff measure, , is a given measure and a given parameter. Regularity of minimizers is a delicate problem. It is known that even if is absolutely continuous with respect to Lebesgue measure, regularity does not hold in general. An interesting question is whether the set of corners, i.e. points where 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 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