  This paper is devoted to the study the $m$-point homogeneity property and the
point homogeneity degree for finite metric spaces. Since the vertex sets of
regular polytopes, as well as of some their generalizations, are homogeneous,
we pay much attention to the study of the homogeneity properties of the vertex
sets of polytopes in Euclidean spaces. Among main results, there is a
classification of polyhedra with all edges of equal length and with 2-point
homogeneous vertex sets. In addition, a significant part of the paper is
devoted to the development of methods and tools for studying the relevant
objects.
