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.