  We study homogeneity aspects of metric spaces in which all triples of
distinct points admit pairwise different distances; such spaces are called
isosceles-free. In particular, we characterize all homogeneous isosceles-free
spaces up to isometry as vector spaces over the two-element field, endowed with
an injective norm. Using isosceles-free decompositions, we provide bounds on
the maximal number of distances in arbitrary homogeneous finite metric spaces.
