Recall the two classical canonical isometric embeddings of a finite metric space X into a Banach space. That is, the Hausdorff–Kuratowsky embedding x → ρ(x, ⋅) into the space of continuous functions on X with the max-norm, and the Kantorovich–Rubinshtein embedding x → δ x (where δ x , is the δ-measure concentrated at x) with the transportation norm. We prove that these embeddings are not equivalent if |X| > 4. Bibliography: 2 titles.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
L. V. Kantorovich and G. Sh. Rubinshtein, “On a space of totally additive functions,” Vestn. Leningr. Univ., 13, No. 7, 52–59 (1958).
J. Melleray, F. V. Petrov, and A. M. Vershik, “Linearly rigid metric spaces and the embedding problem,” Fund. Math., 199, No. 2, 177–194 (2008).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 360, 2008, pp. 153–161.
Rights and permissions
About this article
Cite this article
Zatitskiy, P.B. On the coincidence of the canonical embeddings of a metric space into a Banach space. J Math Sci 158, 853–857 (2009). https://doi.org/10.1007/s10958-009-9422-2
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-009-9422-2