Block realizations of finite metrics and the tight-span construction I: The embedding theorem

Given a finite set XX and a proper metric D:X×X→R≥0D:X×X→R≥0 defined on XX, we show that every block realization of DD can be “embedded” canonically into the tight span T(D)T(D) of DD and characterize the subsets of T(X)T(X) that can be obtained in that way as the “canonical image” of the vertex set of a block realization.