On the coincidence of the canonical embeddings of a metric space into a Banach space

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.