Counterexample to regularity in average-distance problem

The average-distance problem is to find the best way to approximate (or represent) a given measure μ 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 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.