Counterexample to regularity in average-distance problem

Slepčev, Dejan

Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) no. 1, pp. 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

$$E\left(\Sigma \right)=\underset{{ℝ}^d}{\int }d(x,\Sigma )d\mu \left(x\right)+\lambda {\mathscr{H}}^1\left(\Sigma \right)$$

among connected closed sets, Σ, where $\lambda >0$, $d(x,\Sigma )$ is the distance from x to the set Σ, and ${\mathscr{H}}^1$ 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.

[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 | Zbl

[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 | EuDML | Numdam | MR | Zbl

[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

[4] E. Paolini, E. Stepanov, Qualitative properties of maximum distance minimizers and average distance minimizers in ${ℝ}^n$, 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 | Zbl

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

[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 | EuDML | MR | Zbl

[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 | Zbl

[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 | Zbl

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

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

[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