  One can define what it means for a compact manifold with corners to be a
"contractible manifold with contractible faces." Two combinatorially
equivalent, contractible manifolds with contractible faces are diffeomorphic if
and only if their 4-dimensional faces are diffeomorphic. It follows that two
simple convex polytopes are combinatorially equivalent if and only if they are
diffeomorphic as manifolds with corners. On the other hand, by a result of
Akbulut, for each n > 3, there are smooth, contractible n-manifolds with
contractible faces which are combinatorially equivalent but not diffeomorphic.
Applications are given to rigidity questions for reflection groups and smooth
torus actions.
