# Counterexample to regularity in average-distance problem

### Dejan Slepčev

Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, United States

## Abstract

The average-distance problem is to find the best way to approximate (or represent) a given measure *μ* on $R_{d}$ by a one-dimensional object. In the penalized form the problem can be stated as follows: given a finite, compactly supported, positive Borel measure *μ*, minimize

among connected closed sets, *Σ*, where $λ>0$, $d(x,Σ)$ is the distance from *x* to the set *Σ*, and $H_{1}$ is the one-dimensional Hausdorff measure. Here we provide, for any $d⩾2$, an example of a measure *μ* with smooth density, and convex, compact support, such that the global minimizer of the functional is a rectifiable curve which is not $C_{1}$. We also provide a similar example for the constrained form of the average-distance problem.

## Cite this article

Dejan Slepčev, Counterexample to regularity in average-distance problem. Ann. Inst. H. Poincaré Anal. Non Linéaire 31 (2014), no. 1, pp. 169–184

DOI 10.1016/J.ANIHPC.2013.02.004