2013/03/07 by Khek Lun Harold Chao, Chao, Khek Lun Harold
Mathematics · #51F99 #Advanced Algebra and Geometry #Advanced Topology and Set Theory #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Metric Geometry (math.MG) #Point processes and geometric inequalities #math.MG #msc:51F99
paper · pdf · doi:10.48550/arxiv.1303.1801
21 pages, 6 figures. v3: expanded with more results, changed title v2: revised and expanded with new results on symmetry group actions of some regular polytopes
openalex publication_date 2013/03/07 · arxiv created 2014/03/31 · arxiv updated 2014/04/01 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28
In this paper, we first prove the optimal lower bound for Alexandrov angle rigidity of torsion elliptic isometries on any complete CAT(κ) space, which, when attained, leads to an embedded 2-flat in the tangent cone invariant under the induced action of the isometry. Next, we will prove similar result for action of symmetry groups of either a regular orthoplex, a regular hypercube, a regular dodecahedron or a regular icosahedron on a set of points in any complete CAT(κ) space in a way corresponds to the set of vertices of the polytope, the angle made at the circumcenter by any pair of points corresponding to an edge is bounded below by that of the edge in the polytope. As a result, we give a condition for the convex hull of the set of points to be isometric to a corresponding regular polytope in a model space of constant curvature κ.