On $m$-point homogeneous polytopes in Euclidean spaces

TL;DR

This paper investigates the homogeneity properties of finite metric spaces and polytope vertex sets in Euclidean spaces, providing classifications and developing new methods for analyzing these geometric structures.

Contribution

It offers a classification of polyhedra with equal edges and 2-point homogeneous vertex sets, along with new tools for studying homogeneity in Euclidean polytopes.

Findings

Classification of polyhedra with all edges equal and 2-point homogeneous vertices

Development of methods for analyzing homogeneity properties

Insights into the structure of homogeneous vertex sets in Euclidean polytopes

Abstract

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.