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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!

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Joseph, this midsphere theorem is really cool! I have not got from the wiki link though if there is good (comprehensible) reference for a proof of this theorem, is there? –  Dmitri May 21 '11 at 21:38
@Dmitri: I recall there is a lucid discussion in the (masterful) Pach-Agarwal book, Combinatorial Geometry, , but I don't have it with me and cannot consult it at the moment. –  Joseph O'Rourke May 21 '11 at 22:00
@Dmitri: if you read french, you might want to read Y. Colin de Verdiere's Inventiones paper from around 1992. –  Igor Rivin May 22 '11 at 2:59

5 Answers 5

up vote 9 down vote accepted

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...

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@Gil: Thanks so much for your insights! This may explain why my searches have failed: There are no positive results to report. Interesting idea for the 5-polytope extension... –  Joseph O'Rourke May 21 '11 at 22:01
In the introduction of "how to cage an egg", Schramm discusses a little more this dimension counting. On the positive side of things one can substitute the 2-sphere by any smooth strictly convex body. –  Alfredo Hubard Oct 31 '11 at 13:54

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)

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@Igor: Thanks!, especially for the Apollonian article, which reminded me of this: For any $d+1$ mutually tangent spheres in $d$ dimensions, There is a unique sphere through their points of tangency, orthogonal to each of the $d+1$ spheres. –  Joseph O'Rourke May 22 '11 at 17:58
Would that be the circumsphere of the convex hull of the points of tangency? –  Igor Rivin May 23 '11 at 16:21
The apollonian case correspond to the class of polytopes known as "stacked polytopes" which are obtained by gluing simplices along facets. Stacked (d+1)-polytopes lead to apollonian sphere packing in d-dimension defined inductively. For dimension > 3 I suspect that the spheres are not necessarily non overlapping but I am not sure. (Maybe we should relac the nonoverlapping requirement...) –  Gil Kalai May 23 '11 at 18:02
@Gil: You are right, P_6*C_3 is the graph of a stacked polytope, but not sphere packable. –  Hao Chen Sep 3 '12 at 15:05

I recently showed that:

The graph of a stacked $4$-polytope is $3$-ball packable if and only if it does not contain six $4$-cliques sharing a $3$-clique.

While Eppstein, Kuperberg and Ziegler 2003 proved that

No stacked 4-polytope with more than 6 vertices is edge-tangent.

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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.

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Citation: Schulte, Egon. "Analogues of Steinitz’s theorem about noninscribable polytopes." Intuitive geometry (Siófok 1985) (1985): 503-516. –  Joseph O'Rourke Apr 14 at 22:44

In a recent paper of Padrol and me, we studied several generalizations of this problem.

Regarding Q1, Yoav already mentioned Schulte's work, and Gil mentioned that stacked polytopes won't work. In fact,

For any 0 ≤ k ≤ d − 3, there are stacked d-polytopes that are not k-scribable.

But the other side of the story is much more joyful:

Every stacked polytope is (d − 1)-scribable (i.e. circumscribable) and (d − 2)-scribable (i.e. ridge-scribable). Dually, it means that every truncated polytope is edge-scribable and inscribable. In particular, the $1$-skeleton of every truncated polytope is the tangency graph of a sphere packing.

We also studied the weak scribabilities, also mentioned in Schulte's work.

The most interesting generalization (which I would like to make some advertisement here) is the $(i,j)$-scribability problem:

Can we realize every $d$-polytope with all $i$-faces "outside" the sphere and all $j$-faces "intersecting" the sphere, $0\le i \le j \le d-1$?

The case of $i=j$ is just the classical scribability problem. For $i<j$, we constructed examples with no such realization for all but two cases: $j-i=d-2$ for odd $d$, or $j-i=d-1$ for any $d$.

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