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

Abstract

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.