Geodesic
Counterexample to regularity in average-distance problem
Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) no. 1, pp. 169-184

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The average-distance problem is to find the best way to approximate (or represent) a given measure μ on 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 λ>0, d(x,Σ) is the distance from x to the set Σ, and 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.

DOI : 10.1016/j.anihpc.2013.02.004
Classification : 49Q20, 49K10, 49Q10, 05C05, 35B65
Keywords: Average-distance problem, Nonlocal variational problem, Regularity 
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     author = {Slep\v{c}ev, Dejan},
     title = {Counterexample to regularity in average-distance problem},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {169--184},
     publisher = {Elsevier},
     volume = {31},
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     year = {2014},
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Slepčev, Dejan. Counterexample to regularity in average-distance problem. Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) no. 1, pp. 169-184. doi: 10.1016/j.anihpc.2013.02.004

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