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?



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


