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 μ 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 μ, minimize

among connected closed sets, Σ, where $\lambda >0$, $d(x,\Sigma )$ is the distance from x to the set Σ, and ${\mathcal{H}}^1$ 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.