  In this paper, we first prove the optimal lower bound for Alexandrov angle
rigidity of torsion elliptic isometries on any complete CAT($\kappa$) 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($\kappa$) 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 $\kappa$.
