The (remarkable) *midsphere* theorem says that each combinatorial
type of convex polyhedron may be realized by one all of whose edges are
tangent to a sphere
(and the realization is unique if the center of gravity is specified).

Q1.Is there an analogous theorem for 4-polytopes, that each combinatorial type may be realized by a polytope with ridges (or edges?) tangent to a 3-sphere?

Because the proofs of the midsphere theorem rely on the Koebe–Andreev–Thurston circle-packing theorem, a related query is:

Q2.Is there a generalization of the circle-packing theorem to sphere-packing?

Both questions may be generalized to arbitrary dimension.

I suspect the answer to both questions may be *No*,
in which case a pointer would suffice. Thanks!

Combinatorial Geometry, amazon.com/Combinatorial-Geometry-225-nos-Pach/dp/0471588903 , but I don't have it with me and cannot consult it at the moment. $\endgroup$ – Joseph O'Rourke May 21 '11 at 22:00