Geodesic
On the coincidence of the canonical embeddings of a metric space into a Banach space
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamics systems, combinatorial methods. Part XVI, Tome 360 (2008), pp. 153-161 
P. B. Zatitskii. On the coincidence of the canonical embeddings of a metric space into a Banach space. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamics systems, combinatorial methods. Part XVI, Tome 360 (2008), pp. 153-161. http://geodesic.mathdoc.fr/item/ZNSL_2008_360_a6/
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     title = {On the coincidence of the canonical embeddings of a~metric space into {a~Banach} space},
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

Recall the two classical canonical isometric embeddings of a finite metric space $X$ into a Banach space. That is, the Hausdorff–Kuratowsky embedding $x\to\rho(x,\cdot)$ into the space of continuous functions on $X$ with the max-norm, and the Kantorovich–Rubinshtein embedding $x\to\delta_x$ (where $\delta_x$ is the $\delta$-measure concentrated at $x$) with the transportation norm. We prove that these embeddings are not equivalent if $|X|>4$. Bibl. – 2 titles.

[1] L. V. Kantorovich, G. Sh. Rubinshtein,, “On a space of totally additive functions”, Vestn. Leningr. Univ., 13:7 (1958), 52–59 | MR | Zbl

[2] J. Melleray, F. V. Petrov, A. M. Vershik, “Linearly rigid metric spaces and the embedding problem”, Fund. Math., 199:2 (2008), 177–194 | DOI | MR | Zbl