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!
Best Answer
Dear Joe,
As far as I remember all attempts to extend the midsphere theorem and the ball packing theorem for 4-polytopes turned out to be false. I remember discussing it with Oded Schramm and even very simple cases of Q2 like for stacked 4-polytopes or for pyramids over 3-polytopes did not work. Somehow the number of degrees of freedoms for the vertices of 4-polytopes or higher is not sufficient. (And even if you consider special cases where the number of degree of freedoms is fine still the theorems do not extend.)
One possible extension I would be pleased to see is to realize generalized 5-polytopes so all 2-faces are tangent to a sphere, where these generalized gadgets each "edge" is not a steight line edge but you can bend it (say with 4 degrees of freedom). But as much as I will be pleased to see such a reasonable generalization formulated I would immediately guess it is false...