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Generalized Transportation Cost Spaces

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Abstract

The paper is devoted to the geometry of transportation cost spaces and their generalizations introduced by Melleray et al. (Fundam Math 199(2):177–194, 2008). Transportation cost spaces are also known as Arens–Eells, Lipschitz-free, or Wasserstein 1 spaces. In this work, the existence of metric spaces with the following properties is proved: (1) uniformly discrete metric spaces such that transportation cost spaces on them do not contain isometric copies of \(\ell _1\), this result answers a question raised by Cúth and Johanis (Proc Am Math Soc 145(8):3409–3421, 2017); (2) locally finite metric spaces which admit isometric embeddings only into Banach spaces containing isometric copies of \(\ell _1\); (3) metric spaces for which the double-point norm is not a norm. In addition, it is proved that the double-point norm spaces corresponding to trees are close to \(\ell _\infty ^d\) of the corresponding dimension, and that for all finite metric spaces M, except a very special class, the infimum of all seminorms for which the embedding of M into the corresponding seminormed space is isometric, is not a seminorm.

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References

  1. Arens, R.F., Eells Jr., J.: On embedding uniform and topological spaces. Pac. J. Math. 6, 397–403 (1956)

    MathSciNet  MATH  Google Scholar 

  2. Baudier, F., Lancien, G.: Embeddings of locally finite metric spaces into Banach spaces. Proc. Am. Math. Soc. 136, 1029–1033 (2008)

    MathSciNet  MATH  Google Scholar 

  3. Billingsley, P.: Convergence of Probability Measures. Wiley, New York (1968)

    MATH  Google Scholar 

  4. Blumenthal, L.M.: Theory and Applications of Distance Geometry. Clarendon Press, Oxford (1953)

    MATH  Google Scholar 

  5. Cúth, M., Doucha, M.: Lipschitz-free spaces over ultrametric spaces. Mediterr. J. Math. 13(4), 1893–1906 (2016)

    MathSciNet  MATH  Google Scholar 

  6. Cúth, M., Doucha, M., Wojtaszczyk, P.: On the structure of Lipschitz-free spaces. Proc. Am. Math. Soc. 144(9), 3833–3846 (2016)

    MathSciNet  MATH  Google Scholar 

  7. Cúth, M., Johanis, M.: Isometric embedding of \(\ell _1\) into Lipschitz-free spaces and \(\ell _\infty \) into their duals. Proc. Am. Math. Soc. 145(8), 3409–3421 (2017)

    MATH  Google Scholar 

  8. Dalet, A.: Free spaces over some proper metric spaces. Mediterr. J. Math. 12(3), 973–986 (2015)

    MathSciNet  MATH  Google Scholar 

  9. Dilworth, S.J., Kutzarova, D., Ostrovskii, M.I.: Lipschitz-free spaces on finite metric spaces. Canad. J. Math. to appear, ar**v:1807.03814

  10. Dutrieux, Y., Lancien, G.: Isometric embeddings of compact spaces into Banach spaces. J. Funct. Anal. 255(2), 494–501 (2008)

    MathSciNet  MATH  Google Scholar 

  11. Fréchet, M.: Les dimensions d’un ensemble abstrait. Math. Ann. 68(2), 145–168 (1910)

    MathSciNet  MATH  Google Scholar 

  12. Garling, D.J.H.: Analysis on Polish Spaces and an Introduction to Optimal Transportation. London Mathematical Society Student Texts, vol. 89. Cambridge University Press, Cambridge (2018)

    Google Scholar 

  13. Godard, A.: Tree metrics and their Lipschitz-free spaces. Proc. Am. Math. Soc. 138(12), 4311–4320 (2010)

    MathSciNet  MATH  Google Scholar 

  14. Godefroy, G., Kalton, N.J.: Lipschitz-free Banach spaces. Stud. Math. 159(1), 121–141 (2003)

    MathSciNet  MATH  Google Scholar 

  15. Kadets, V.M.: Lipschitz map**s of metric spaces (Russian). Izv. Vyssh. Uchebn. Zaved. Mat. 1, 30–34 (1985). (English transl.: Soviet Math. (Iz. VUZ), 29 (1985), no. 1, 36–41)

    Google Scholar 

  16. Kalton, N.J., Lancien, G.: Best constants for Lipschitz embeddings of metric spaces into \(c_0\). Fundam. Math. 199, 249–272 (2008)

    MATH  Google Scholar 

  17. Kantorovich, L.V.: On mass transportation (Russian). Doklady Akad. Nauk SSSR (N.S.) 37, 199–201 (1942). (English transl.: J. Math. Sci. (N. Y.), 133 (2006), no. 4, 1381–1382)

    Google Scholar 

  18. Kantorovich, L.V.: Mathematical-Economic Articles. Selected Works (Russian). Nauka, Novosibirsk (2011)

    MATH  Google Scholar 

  19. Kantorovich, L.V., Akilov, G.P.: Functional Analysis. Translated from the Russian by Howard L. Silcock, 2nd edn. Pergamon Press, Oxford (1982)

    MATH  Google Scholar 

  20. Kantorovich, L.V., Akilov, G.P.: Functional Analysis (Russian), 3rd edn. Nauka, Moscow (1984)

    MATH  Google Scholar 

  21. Kantorovich, L.V., Gavurin, M.K.: Application of mathematical methods in the analysis of cargo flows (Russian). In: Problems of Improving of Transport Efficiency. pp. 110–138. USSR Academy of Sciences Publishers, Moscow (1949)

  22. Kantorovich, L.V., Rubinstein, G.S.: On a functional space and certain extremum problems (Russian). Dokl. Akad. Nauk SSSR (N.S.) 115, 1058–1061 (1957)

    MathSciNet  MATH  Google Scholar 

  23. Kantorovich, L.V., Rubinstein, G.S.: On a space of completely additive functions (Russian). Vestnik Leningrad. Univ. 13(7), 52–59 (1958)

    MathSciNet  Google Scholar 

  24. Kusraev, A.G., Kutateladze, S.S.: Kantorovich spaces and optimization. Teor. Predst. Din. Sist. Komb. i Algoritm. Metody. 11, 138–149 (2004). (English translation in J. Math. Sci. (N.Y.) 133 (2006), no. 4, pp. 1449–1455)

    Google Scholar 

  25. Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces. Lecture Notes in Mathematics, vol. 338. Springer, Berlin (1973)

    MATH  Google Scholar 

  26. Markov, A.: On free topological groups. Doklady Akad. Nauk SSSR 31, 299–301 (1941)

    MathSciNet  Google Scholar 

  27. Markov, A.: On free topological groups (Russian). Izvestiya Akad. Nauk SSSR 9, 3–64 (1945). (; English translation in: Translations, Ser. 1, Vol. 8: Topology and topological algebra. American Mathematical Society, Providence, R.I. 1962, pp. 195–272)

    Google Scholar 

  28. Maurey, B.: Type, Cotype and \(K\)-Convexity. Handbook of the Geometry of Banach Spaces, vol. 2, pp. 1299–1332. North-Holland, Amsterdam (2003)

    MATH  Google Scholar 

  29. Melleray, J., Petrov, F.V., Vershik, A.M.: Linearly rigid metric spaces and the embedding problem. Fundam. Math. 199(2), 177–194 (2008)

    MathSciNet  MATH  Google Scholar 

  30. Michael, E.: A short proof of the Arens–Eells embedding theorem. Proc. Am. Math. Soc. 15, 415–416 (1964)

    MathSciNet  MATH  Google Scholar 

  31. Naor, A.: Metric dimension reduction: a snapshot of the Ribe program. Proc. Int. Cong. Math. 1, 759–838 (2018)

    Google Scholar 

  32. Ostrovska, S., Ostrovskii, M.I.: Distortion in the finite determination result for embeddings of locally finite metric spaces into Banach spaces. Glasg. Math. J. 61(1), 33–47 (2019)

    MathSciNet  MATH  Google Scholar 

  33. Ostrovska, S., Ostrovskii, M.I.: On embeddings of locally finite metric spaces into \(\ell _p\). J. Math. Anal. Appl. 474, 666–673 (2019)

    MathSciNet  MATH  Google Scholar 

  34. Ostrovskii, M.I.: Embeddability of locally finite metric spaces into Banach spaces is finitely determined. Proc. Am. Math. Soc. 140, 2721–2730 (2012)

    MathSciNet  MATH  Google Scholar 

  35. Ostrovskii, M.I.: Metric Embeddings: Bilipschitz and Coarse Embeddings into Banach Spaces. de Gruyter Studies in Mathematics, 49. Walter de Gruyter & Co., Berlin (2013)

    Google Scholar 

  36. Ostrovskii, M.I.: Different forms of metric characterizations of classes of Banach spaces. Houst. J. Math. 39(3), 889–906 (2013)

    MathSciNet  MATH  Google Scholar 

  37. Pestov, V.G.: Free Banach spaces and representations of topological groups (Russian). Funktsional. Anal. i Prilozhen. 20(1), 81–82 (1986). (English transl.: Funct. Anal. Appl.20 (1986), 70–72)

    MathSciNet  Google Scholar 

  38. Rachev, S.T., Rüschendorf, L.: Mass Transportation Problems: Theory. Probability and its Applications, vol. 1. Springer, New York (1998)

    MATH  Google Scholar 

  39. Rubner, Y., Tomasi, C., Guibas, L.J.: A metric for distributions with applications to image databases. In: Proceedings ICCV 1998. pp. 59–66. https://doi.org/10.1109/ICCV.1998.710701

  40. Shimrat, M.: Embedding in homogeneous spaces. Q. J. Math. Oxford Ser. (2) 5, 304–311 (1954)

    MathSciNet  MATH  Google Scholar 

  41. Vershik, A.M.: The Kantorovich metric: the initial history and little-known applications (Russian). Teor. Predst. Din. Sist. Komb. i Algoritm. Metody. 11, 69–85 (2004). (translation in J. Math. Sci. (N.Y.) 133 (2006), no. 4, 1410–1417)

    Google Scholar 

  42. Vershik, A.M.: Long history of the Monge–Kantorovich transportation problem. Math. Intell. 35(4), 1–9 (2013)

    MathSciNet  MATH  Google Scholar 

  43. Vasershtein, L.N.: Markov processes over denumerable products of spaces describing large system of automata. Probl. Inf. Transm. 5(3), 47–52 (1969). (translated from: Problemy Peredachi Informatsii5 (1969), no. 3, 64–72)

    MathSciNet  Google Scholar 

  44. Villani, C.: Topics in Optimal Transportation. Graduate Studies in Mathematics, vol. 58. American Mathematical Society, Providence (2003)

    MATH  Google Scholar 

  45. Villani, C.: Optimal Transport. Old and New. Grundlehren der Mathematischen Wissenschaften, vol. 338. Springer, Berlin (2009)

    Google Scholar 

  46. Weaver, N.: Lipschitz Algebras. World Scientific Publishing Co., Inc., River Edge (1999)

    MATH  Google Scholar 

  47. Weaver, N.: Lipschitz Algebras, vol. 2. World Scientific Publishing Co. Pte. Ltd., Hackensack (2018)

    MATH  Google Scholar 

  48. Zatitskii, P.B.: On the coincidence of the canonical embeddings of a metric space into a Banach space. J. Math. Sci. 158(6), 853–857 (2009). (translation from Zap. Nauchn. Semin. POMI, 360 (2008), 153–161)

    MathSciNet  Google Scholar 

  49. Zatitskii, P.B.: Canonical embeddings of compact metric spaces. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. 18, 40–46 (2010). (translation in J. Math. Sci. (N.Y.) 174 (2011), no. 1, 19–22)

    MathSciNet  Google Scholar 

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Acknowledgements

The second-named author gratefully acknowledges the support by National Science Foundation Grant NSF DMS-1700176. We would like to thank the referee for the valuable suggestions and corrections.

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  1. Department of Mathematics, Atilim University, 06830, Incek, Ankara, Turkey

    Sofiya Ostrovska

  2. Department of Mathematics and Computer Science, St. John’s University, 8000 Utopia Parkway, Queens, NY, 11439, USA

    Mikhail I. Ostrovskii

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  1. Sofiya Ostrovska
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Correspondence to Mikhail I. Ostrovskii.

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Ostrovska, S., Ostrovskii, M.I. Generalized Transportation Cost Spaces. Mediterr. J. Math. 16, 157 (2019). https://doi.org/10.1007/s00009-019-1433-8

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