On $m$-point homogeneous polyhedra in $3$-dimensional Euclidean space

TL;DR

This paper classifies 2-point and 3-point homogeneous polyhedra in three-dimensional Euclidean space, advancing understanding of symmetry properties of polytope vertex sets.

Contribution

It provides the first complete classification of 2-point and 3-point homogeneous polyhedra in 3D Euclidean space.

Findings

Classified all 2-point homogeneous polyhedra in D

Classified all 3-point homogeneous polyhedra in D

Established criteria for m-point homogeneity in polyhedra

Abstract

This paper is devoted to the study of the $m$-point homogeneity property for the vertex sets of polytopes in Euclidean spaces. In particular, we present the classifications of $2$-point and $3$-point homogeneous polyhedra in $\mathbb{R}^3$.