Touching Simplices and Polytopes: Perles’ argument Joseph Zaks (1984), picture taken by Ludwig Danzer (OberWolfach photo collection)

The story I am going to tell here was told in several places, but it might be new to some readers and I will mention my own angle, recall and raise some questions  and provide some comments. You can read about it here: Martin Aigner and Günter Ziegler, Proofs from the Book Chapter 13 (freely available on Springer links), and also in Bela Bollobas, The Art of Mathematics – Coffee Time in Memphis.

Can you find  non-overlapping $d$-dimensional polytopes $P_1,P_2,\dots ,P_k$ be, in $\mathbb R^d$, such that each pair of polytopes have $(d-1)$-dimensional intersection, and such that $P_i$ has $f_i$ facets? In two dimensions $k \le 4$ (because $K_5$ is not planar) but in higher dimensions things are not clear. We call two d-polytopes in $\mathbb R^d$ “touching” if they have $(d-1)$-dimensional intersection (no more, no less).

A special case of this problem deals with the case that all the $P_i$s are simplices (thus having $d+1$ facets each) and then the question is how large $k$ can be? The problem goes back to a 1956 paper by Bagemihl. A famous conjecture  by Bagemihl (space) and Baston (all dimensions) asserts that $k \le 2^d$. Joseph Zaks gave a beautiful example of $2^d$ simplices in $\mathbb R^d$ with this property. Baston proved that in space $k \le 9$ and, based on his work and a great deal of extra effort, Joseph Zaks verified Bagemihl’s original conjecture! He thus proved that in space $k \le 8$. The conjecture in higher dimensions is wide open.

Let me show you a very simple proof by Micha A. Perles that $k \le 2^{d+1}$.

Let n be the overall number of hyperplanes supporting the facets of the polytopes and let $H_1,H_2, \dots , H_n$ be these supporting hyperplanes. For each hyperplane $H_i$ we denote the two closed half spaces determined by it by $H_i^+$ and $H_i^-$. Now let us suppose that in our family of polytopes there are $r_j$ polytopes with $j$ facets, $j=d+1,d+2,\dots$.

Theorem:

(*) $\sum r_j 2^{-j} \le 1$.

In particular, if all polytopes are simplices we get that $k\le 2^{d+1}$.

Proof (Micha A. Perles):  Consider a matrix $M=(m_{ij})$ whose rows correspond to the $k$ polytopes and whose columns correspond to the $n$ hyperplanes.

The entries are: $m_{ij}=1$ if $H_j$ contains a facet of polytope $P_i$, and $P_i \subset H_i^+$. $m_{ij}=-1$ if $H_j$ contains a facet of polytope $P_i$, and $P_i \subset H_i^-$.

Otherwise, $m_{ij}=0$.

Now replace each row of the matrix by new $2^{n-f_i}$ rows by replacing every zero entry by a $\pm 1$ entry in all possible ways.  (Recall that polytope $P_i$ had $f_i$ facets.)

Claim: All the new rows are distinct.

Proof: This is clear for two new rows coming from the same original row. Given two rows which correspond to distinct polytopes $P_e$ and $P_f$, since these polytopes are touching, in the original rows there is a column $j$ for which $m_{ej}=1$ and $m_{fj}=-1$ or the other way around. These pair of distinct entries remain unchanged when zeroes are replaced by $\pm 1$. Ahla!

Returning to the theorem, since there are altogether $2^n$ vectors of $\pm 1$ entries we get that $\sum _{i=1}^k 2^{n-f_i} = \sum_{j \ge d+1} r_j 2^{n-j} \le 2^n$. Dividing by $2^n$ gives us the theorem.  Sababa!

Of course, we can ask if a stronger inequality is valid and propose that

Conjecture  (**): $\sum r_j 2^{-j} \le 1/2$.

Another question is when (for $d \ge 3$) can we realize “ $r$-vectors” obeying the strong inequality (**) by a family of touching polytopes.

Twelve quick remarks:

1. Perles’ argument is close in spirit to a proof technique in extremal combinatorics that we saw, for example, in the proof of Sperner’s theorem on antichains and in the proof of the “two family theorems“. (Here and below you can find links to earlier blog posts, but not to the precise location in those posts.)
2. Inequality (*) resembles the LYM inequality.
3. Zaks’ example of $2^d$ touching simplices in $\mathbb R^d$ is beautiful. See his paper Neighborly family of $2^d$ simplices in $E^d$. (Or here.)
4. Let us call a facet of a polytope in the family “free” if it does not touch any other polytope. Zaks made the following conjecture: In every family of touching $d-$simplices ( $d-$polytopes) in $\mathbb R^d$, there is a free facet (to at least one of the polytopes.)
5. In a family of touching simplices if every simplex has a free face then $k\le 2^d$. (More generally, for touching polytopes with this property (**) holds.)
6.  It would be very nice to prove the $2^d$ upper bounds for families of touching simplices (and inequality (**) for families of touching polytopes) such that in the family and in every subfamily at least one simplex has a free facet.
7. Related to comments 1 and 6. Is there an algebraic/topological proof of the known inequality which will enable to use Zaks’ free-facet conjecture? Possible role model: Lovasz’ algebraic proof (and geometric extension) of the two-family theorem.
8. Two $d$-polytopes $P,Q$ in $\mathbb R^d$ are weakly touching if there is a hyperplane containing a facet of both that (weakly) separates them.
9. Perles’ proof and inequality (*) extend for families of weakly touching polytopes. It may be the case that the conjectures of Bagemihl and Baston and inequality (**) extends for weakly touching polytopes.
10. Zaks’ free-facet conjecture does not extend for weakly touching simplices.
11. In my first meeting with Gunter Ziegler in 1984 we talked about this problem. (We had both worked on it previously).
12. We use here again the convention of ending proofs with the words Walla and Sababa. The endings Ahla, Ashkara (for trivial proofs), and Fadiha (for wrong proofs) can also be used.

A   fantasy for attacking the problem based on points 4,6,7 above is given in the following figure Coxeter’s MathSciNet review of Baston’s Book

Baston, V. J. D. , Some properties of polyhedra in Euclidean space.
International Series of Monographs in Pure and Applied Mathematics, Vol. 71 Pergamon Press, Oxford-Edinburgh-New York 1965 xi+209 pp.

F. Bagemihl [Amer. Math. Monthly 63 (1956), 328–329; MR0077132] considered a set of n tetrahedra (in three-dimensional space) which fit together without overlapping so that every two of them have a common boundary (part of a face) of positive area. After proving that n can be 8, and that n≤17, he conjectured that 8 is the greatest possible value of n. The author, in the first chapter, reaches the same conclusions independently, representing the arrangement by a matrix of n rows whose entries are all 1, 0 or -1. In the remaining nine chapters he reduces the upper bound from 17 to 9, still leaving for future research the possibility of an arrangement with n=9.

The presentation is enlivened by colorful nomenclature; for instance, one reads, on page 181, “As S3 contains five dotres and no treuns there are ten tripts in the matrix. As there are four naiks…” Although such passages are less poetic then “All mimsy were the borogoves, and the mome raths outgrabe”, there is again the comforting thought that each unfamiliar word is precisely defined in another part of the book. Moreover, the index reassures us that there are only 24 such words, including Lewis Carroll’s old friend, the dodo.

Reviewed by H. S. M. Coxeter