There's a famous theorem of Cauchy (proving a claim of Euclid it seems) that two convex polyhedra with the same shape faces, assembled the same way, are in fact congruent. In other words, if you're building a polyhedron from its faces, it may be floppy when you're part way through, but when you finish, it is rigid.
Is the same true in higher dimensions? E.g., in R^4, if I assemble a convex polytope from certain 3-cells (polyhedra, tetrahedra if you like) of fixed shape, is the result always rigid?
I assume whatever results there are along these lines fall under Alexandrov's theory of mixed volumes, etc, for convex polytopes. Unfortunately, our library only had Alexandrov in Russian, so I figured I'd ask the net if anyone knows the statement of whatever analogous theorem there might be in higher dimensions.