We talk about the following minimization problem:
where Ω is an open subset of \( {\mathbb{R}^2} \), μ is a probability measure, and the minimum is taken over all sets \( \Sigma \subset \overline \Omega \) such that Σ is compact, connected, and \( {\mathcal{H}^1}\left( \Sigma \right) \leq {\alpha_0} \) for a given positive constant α 0. Bibliography: 21 titles.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
L. Ambrosio, N. Fusco, and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. The Clarendon Press, Oxford Univ. Press, New York (2000).
L. Ambrosio and P. Tilli, Topics on Analysis in Metric Spaces, Oxford Univ. Press, Oxford (2004).
D. Bucur, I. Fragalà, and J. Lamboley, “Optimal convex shapes for concave functionals,” to appear in ESAIM COCV.
G. Buttazzo, E. Mainini, and E. Stepanov, “Stationary configurations for the average distance functional and related problems,” Control Cybernet., 38, No. 4A, 1107–1130 (2009).
G. Buttazzo, E. Oudet, and E. Stepanov, “Optimal transportation problems with free Dirichlet regions,” in: Variational Methods for Discontinuous Structures, 51, Progr. Nonlinear Differential Equations Appl., Birkhäuser, Basel (2002), pp. 41–65.
G. Buttazzo, A. Pratelli, S. Solimini, and E. Stepanov, Optimal Urban Networks via Mass Transportation, Lect. Notes Math., 1961, Springer-Verlag, Berlin (2009).
G. Buttazzo and F. Santambrogio, “Asymptotical compliance optimization for connected networks,” Netw. Heterog. Media, 2, No. 4, 761–777 (2007).
G. Buttazzo and 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, 631–678 (2003).
G. David and S. Semmes, Analysis of and on Uniformly Rectifiable Sets, Amer. Math. Soc., Providence, Rhode Island (1993).
K. J. Falconer, The Geometry of Fractal Sets, Cambridge Univ. Press, Cambridge (1986).
S. Jaffard, “Formalisme multifractal pour les fonctions,” C. R. Acad. Sci. Paris Sér. I Math., 317, No. 8, 745–750 (1993).
S. Jaffard, “Old friends revisited: the multifractal nature of some classical functions,” J. Fourier Anal. Appl., 3, No. 1, 1–22 (1997)
S. Jaffard, “Multifractal functions: recent advances and open problems,” Bull. Soc. Roy. Sci. Liège, 73, No. 2–3, 129–153 (2004).
A. Lemenant, “About the regularity of average distance minimizers in \( {\mathbb{R}^2} \),” J. Conves Anal., 18, No. 4, 949–981 (2011).
A. Lemenant and E. Mainini, “On convex sets that minimize the average distance,” to appear in ESAIM COCV.
F. Morgan, “(M, ϵ, δ)-minimal curve regularity,” Proc. Amer. Math. Soc., 120, No. 3, 677–686 (1994).
S. Mosconi and P. Tilli, “Γ-convergence for the irrigation problem,” J. Convex Anal., 12, No. 1, 145–158 (2005).
E. Paolini and E. Stepanov, “Qualitative properties of maximum distance minimizers and average distance minimizers in \( {\mathbb{R}^n} \),” J. Math. Sci. (N. Y.), 122, No. 3, 3290–3309 (2004).
F. Santambrogio and P. Tilli, “Blow-up of optimal sets in the irrigation problem,” J. Geom. Anal., 15, No. 2, 343–362 (2005).
F. Stepanov, “Partial geometric regularity of some optimal connected transportation networks,” J. Math. Sci. (N. Y.), 132, No. 4, 522–552 (2006).
P. Tilli, “Some explicit examples of minimizers for the irrigation problem,” J. Convex Anal., 17, No. 2, 583–595 (2010).
Author information
Authors and Affiliations
Corresponding author
Additional information
Published in Zapiski Nauchnykh Seminarov POMI, Vol. 390, 2011, pp. 117–146.
Rights and permissions
About this article
Cite this article
Lemenant, A. A presentation of the average distance minimizing problem. J Math Sci 181, 820–836 (2012). https://doi.org/10.1007/s10958-012-0717-3
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-012-0717-3