Counterexample to regularity in average-distance problem
Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 1, p. 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 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)= d d(x,Σ)dμ(x)+λ 1 (Σ) among connected closed sets, Σ, where λ>0, d(x,Σ) is the distance from x to the set Σ, and 1 is the one-dimensional Hausdorff measure. Here we provide, for any d2, 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/

[1] G. Buttazzo, E. Oudet, E. Stepanov, Optimal transportation problems with free Dirichlet regions, Variational Methods for Discontinuous Structures, Progr. Nonlinear Differential Equations Appl. vol. 51, Birkhäuser, Basel (2002), 41-65 | MR 2197837 | Zbl 1055.49029

[2] G. Buttazzo, E. Stepanov, Optimal transportation networks as free Dirichlet regions for the Monge–Kantorovich problem, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 2 no. 4 (2003), 631-678 | Numdam | MR 2040639 | Zbl 1127.49031

[3] A. Lemenant, A presentation of the average distance minimizing problem, Teoriya Predstavlenii, Dinamicheskie Sistemy, Kombinatornye Metody. XX Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 390 (2011), 117-146, http://dx.doi.org/10.1007/s10958-012-0717-3 | MR 2870232

[4] E. Paolini, E. Stepanov, Qualitative properties of maximum distance minimizers and average distance minimizers in n , J. Math. Sci. (N. Y.) 122 no. 3 (2004), 3290-3309, http://dx.doi.org/10.1023/B:JOTH.0000031022.10122.f5

[5] P. Tilli, Some explicit examples of minimizers for the irrigation problem, J. Convex Anal. 17 no. 2 (2010), 583-595 | MR 2675664 | Zbl 1191.49047

[6] A. Lemenant, About the regularity of average distance minimizers in 2 , J. Convex Anal. 18 no. 4 (2011), 949-981 | MR 2917861 | Zbl 1238.49054

[7] G. Buttazzo, E. Mainini, E. Stepanov, Stationary configurations for the average distance functional and related problems, Control Cybernet. 38 no. 4A (2009), 1107-1130 | MR 2779113 | Zbl 1239.49029

[8] F. Santambrogio, P. Tilli, Blow-up of optimal sets in the irrigation problem, J. Geom. Anal. 15 no. 2 (2005), 343-362, http://dx.doi.org/10.1007/BF02922199 | MR 2152486 | Zbl 1115.49029

[9] L. Ambrosio, N. Fusco, D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Math. Monogr., The Clarendon Press, Oxford University Press, New York (2000) | MR 1857292 | Zbl 0957.49001

[10] G. Leoni, A First Course in Sobolev Spaces, Grad. Stud. Math. vol. 105, American Mathematical Society, Providence, RI (2009) | MR 2527916 | Zbl 1180.46001

[11] E.N. Gilbert, H.O. Pollak, Steiner minimal trees, SIAM J. Appl. Math. 16 (1968), 1-29 | MR 223269 | Zbl 0159.22001

[12] F.K. Hwang, D.S. Richards, P. Winter, The Steiner Tree Problem, Ann. Discrete Math. vol. 53, North-Holland Publishing Co., Amsterdam (1992) | MR 1192785