Linearly rigid metric spaces and Kantorovich type norms

We introduce and study the class of linearly rigid metric spaces; these are the spaces that admit a unique, up to isometry, linearly dense isometric embedding into a Banach space. The first nontrivial example of such a space was given by R. Holmes; he proved that the universal Urysohn space has this property. We give a criterion of linear rigidity of a metric space, which allows us to give a simple proof of the linear rigidity of the Urysohn space and other metric spaces. We relate these questions to the general theory of norms and metrics in spaces of measures on a metric space, and introduce the notion of a Banach norm compatible with a given metric; among these norms, the Kantorovich–Rubinshtein transportation norm is the maximal one, and the unit ball in this metric has a direct geometric description in the spirit of root polytopes.