On the Geometry of Finite Homogeneous Subsets of Euclidean Spaces

This survey is devoted to results recently obtained on finite homogeneous metric spaces. One of the main subjects of discussion is the classification of regular and semiregular polytopes in Euclidean spaces by whether or not their vertex sets have the normal homogeneity property or the Clifford–Wolf homogeneity property. Every finite homogeneous metric subspace of a Euclidean space represents the vertex set of a compact convex polytope whose isometry group is transitive on the set of vertices and with all these vertices lying on some sphere. Consequently, the study of such subsets is closely related to the theory of convex polytopes in Euclidean spaces. Normal homogeneity and the Clifford–Wolf homogeneity describe stronger properties than homogeneity. Therefore, it is natural to first check the presence of these properties for the vertex sets of regular and semiregular polytopes. The second part of the survey is devoted to the study of the m-point homogeneity property and the point homogeneity degree for finite metric spaces. We discuss some recent results, in particular, the classification of polyhedra with all edges of equal length and with 2-point homogeneous vertex sets. In addition to the classification results, the paper contains a description of the main tools for the study of the relevant objects.