Date of Original Version
This is the author’s version of a work that was accepted for publication. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version is available at http://dx.doi.org/10.1016/j.anihpc.2013.02.004
Abstract or Description
The average-distance problem is to find the best way to approximate (or represent) a given measure μ on RdRd 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 E(Σ)=∫Rdd(x,Σ)dμ(x)+λH1(Σ)
among connected closed sets, Σ , where λ>0λ>0, d(x,Σ)d(x,Σ) is the distance from x to the set Σ , and H1H1 is the one-dimensional Hausdorff measure. Here we provide, for anyd⩾2d⩾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 C1C1. We also provide a similar example for the constrained form of the average-distance problem.
Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 31, 1, 169-184.