Is there a midsphere theorem for 4-polytopes?

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...

The results do not generalize, and very little is known. You might, however, want to take a look at:

MR1393382 (97k:52022) Cooper, Daryl(1-UCSB); Rivin, Igor(4-WARW-MI) Combinatorial scalar curvature and rigidity of ball packings. Math. Res. Lett. 3 (1996), no. 1, 51–60. 52C15 (57M50)

You might also want to take a look at:

MR2183490 (2009a:11090c) Graham, Ronald L.(1-UCSD-CS); Lagarias, Jeffrey C.(1-MI); Mallows, Colin L.; Wilks, Allan R.(1-ATT3); Yan, Catherine H.(1-TXAM) Apollonian circle packings: geometry and group theory. III. Higher dimensions. (English summary) Discrete Comput. Geom. 35 (2006), no. 1, 37–72. 11E57 (11H55 52C26)

In the paper "Analogues of Steinitz's theorem about non-inscribable polytopes" by E. Schulte, which comes out of the collection "Intuitive Geometry" from 1987, the author seems to prove a negative result for all cases beside the one covered by the midsphere theorem.

The author defines: "Let $d$ and $m$ be natural numbers with $d\ge2$ and $0\le m\le d-1$. A convex $d$-polytope $P$ is called $(m,d)$-scribable ... if there is an isomorphic copy $P'$ of $P$ such that all faces of $P'$ of dimension $m$ are tangent to some Euclidean $(d-1)$-sphere $S$." Isomorphic seems to mean of the same combinatorial type.

Theorem 3 is: "Let $d\ge3$, $0\le m\le d-1$, and $(m,d)\neq(1,3)$. Then, there are infinitely many $(m,d)$-nonscribable convex $d$-polytopes."

In a note added in proof, the author says that reportedly, P. McMullen also obtained some of the same results independently.