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

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 $E\left(\Sigma \right)=\underset{{ℝ}^{d}}{\int }d\left(x,\Sigma \right)\phantom{\rule{0.166667em}{0ex}}d\mu \left(x\right)+\lambda {ℋ}^{1}\left(\Sigma \right)$ among connected closed sets, Σ, where $\lambda >0$, $d\left(x,\Sigma \right)$ 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 : https://doi.org/10.1016/j.anihpc.2013.02.004
Classification:  49Q20,  49K10,  49Q10,  05C05,  35B65
Keywords: Average-distance problem, Nonlocal variational problem, Regularity
@article{AIHPC_2014__31_1_169_0,
author = {Slep\v cev, Dejan},
title = {Counterexample to regularity in average-distance problem},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
publisher = {Elsevier},
volume = {31},
number = {1},
year = {2014},
pages = {169-184},
doi = {10.1016/j.anihpc.2013.02.004},
zbl = {1286.49055},
mrnumber = {3165284},
language = {en},
url = {http://www.numdam.org/item/AIHPC_2014__31_1_169_0}
}

Slepčev, Dejan. Counterexample to regularity in average-distance problem. Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 1, pp. 169-184. doi : 10.1016/j.anihpc.2013.02.004. http://www.numdam.org/item/AIHPC_2014__31_1_169_0/

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