Counterexample to regularity in average-distance problem

  • Dejan Slepčev

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


The average-distance problem is to find the best way to approximate (or represent) a given measure μ on 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 , is the distance from x to the set Σ, and is the one-dimensional Hausdorff measure. Here we provide, for any , 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 . 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