Counterexample to regularity in average-distance problem

<p>The average-distance problem is to find the best way to approximate (or represent) a given measure <em>μ </em> on R^dRd by a one-dimensional object. In the penalized form the problem can be stated as follows: given a finite, compactly supported, positive Borel measure <em>μ</em>, minimize</p> E(Σ)=∫Rdd(x,Σ)dμ(x)+λH1(Σ)<br> <p>among connected closed sets, <em>Σ </em>, where λ>0λ>0, d(x,Σ)d(x,Σ) is the distance from <em>x</em> to the set <em>Σ </em>, and H¹H1 is the one-dimensional Hausdorff measure. Here we provide, for anyd⩾2d⩾2, an example of a measure <em>μ </em> 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.</p>