Date: Jun 16, 1994 6:51 PM
Author: John Sullivan
Subject: Rigidity of Convex Polytopes
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.

Thanks,

John Sullivan