# Is a polytope with vertices on a sphere and all edges of same length already rigid?

Let's say $$P\subset\Bbb R^d$$ is some convex polytope with the following two properties:

1. all vertices are on a common sphere.
2. all edges are of the same length.

I suspect that such a polytope is already rigid, i.e. there is (up to scaling and rotation) only a single way to realize it geometrically. Is this true? What if we only look at uniform polytopes?

I guess you also fix the combinatorial structure. Then yes. Induct in dimension with obvious base $$d\leqslant 2$$. By induction proposition the facets are fixed. By Cauchy - - Alexandrov rigidity theorem the whole polytope also is fixed.
• Thank you very much. Do you have any source for the rigidity theorem in dimension $d>3$? Also, is there anything obvious to say if I do not fix the combinatorial type, but just the edge-graph? – M. Winter Mar 10 '19 at 14:23