Counterexample to regularity in average-distance problem

Abstract

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

among connected closed sets, $\Sigma$, where $\lambda >0$, $d(x,\Sigma )$ is the distance from $x$ to the set $\Sigma$, and ${\mathcal{H}}^1$ is the one-dimensional Hausdorff measure. Here we provide, for any $d⩾2$, an example of a measure $\mu$ 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.