Abstract
In this article, we study the geodesic problem in a generalized metric space, in which the distance function satisfies a relaxed triangle inequality d(x,y)≤σ(d(x,z)+d(z,y)) for some constant σ≥1, rather than the usual triangle inequality. Such a space is called a quasimetric space. We show that many well-known results in metric spaces (e.g. Ascoli-Arzelà theorem) still hold in quasimetric spaces. Moreover, we explore conditions under which a quasimetric will induce an intrinsic metric. As an example, we introduce a family of quasimetrics on the space of atomic probability measures. The associated intrinsic metrics induced by these quasimetrics coincide with the d α metric studied early in the study of branching structures arisen in ramified optimal transportation. An optimal transport path between two atomic probability measures typically has a “tree shaped” branching structure. Here, we show that these optimal transport paths turn out to be geodesics in these intrinsic metric spaces.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Ambrosio, L., Paolo, T.: Topics on Analysis in Metric Spaces. Oxford University Press, Oxford (2004)
Brancolini, A., Buttazzo, G., Santambrogio, F.: Path functions over Wasserstein spaces. J. Eur. Math. Soc. 8(3), 415–434 (2006)
Bernot, M., Caselles, V., Morel, J.: Traffic plans. Publ. Mat. 49(2), 417–451 (2005)
De Pauw, T., Hardt, R.: Size minimization and approximating problems. Calc. Var. Partial Differ. Equ. 17, 405–442 (2003)
Gilbert, E.N.: Minimum cost communication networks. Bell Syst. Technol. J. 46, 2209–2227 (1967)
Heinonen, J.: Lectures on Analysis on Metric Spaces. Springer, New York (2001)
Maddalena, F., Solimini, S., Morel, J.M.: A variational model of irrigation patterns. Interfaces Free Bound. 5(4), 391–416 (2003)
Fagin, R., Kumar, R., Sivakumar, D.: Comparing Top k Lists. SIAM J. Discrete Math. 17(1), 134–160 (2003)
Hunter, J., Nachtergaele, B.: Applied Analysis. World Scientific, Singapore (2001)
Xia, Q.: Optimal paths related to transport problems. Commun. Contemp. Math. 5(2), 251–279 (2003)
Xia, Q.: Interior regularity of optimal transport paths. Calc. Var. Partial Differ. Equ. 20(3), 283–299 (2004)
Xia, Q.: Boundary regularity of optimal transport paths. Preprint
Xia, Q.: The formation of a tree leaf. ESAIM Control Optim. Calc. Var. 13(2), 359–377 (2007)
Xia, Q.: An application of optimal transport paths to urban transport networks. Discrete Contin. Dyn. Syst. 2005, 904–910 (2005)
Xia, Q.: Numerical simulations of optimal transport paths. Preprint
Author information
Authors and Affiliations
Corresponding author
Additional information
This work is supported by an NSF grant DMS-0710714.
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License ( https://creativecommons.org/licenses/by-nc/2.0 ), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Xia, Q. The Geodesic Problem in Quasimetric Spaces. J Geom Anal 19, 452–479 (2009). https://doi.org/10.1007/s12220-008-9065-4
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12220-008-9065-4