Counterexample to regularity in average-distance problem

Slepčev, Dejan

Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014), p. 169-184 / Harvested from Numdam

Résumé

The average-distance problem is to find the best way to approximate (or represent) a given measure μ on $ {\mathbb{R}}^{d}$ 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)=∫ ℝ d d(x,Σ)dμ(x)+λℋ 1 (Σ) among connected closed sets, Σ, where $ \lambda >0$, $ d(x,\Sigma )$ is the distance from x to the set Σ, and $ {\mathcal{H}}^{1}$ is the one-dimensional Hausdorff measure. Here we provide, for any $ d⩾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 $ {C}^{1}$. We also provide a similar example for the constrained form of the average-distance problem.

Détail

Publié le : 2014-01-01

MR 3165284 | Zbl 1286.49055

DOI : https://doi.org/10.1016/j.anihpc.2013.02.004
Classification:  49Q20,  49K10,  49Q10,  05C05,  35B65

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