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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.


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Published In

Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 31, 1, 169-184.