proofs

88 Touching simplices
For our example, this matrix C is a (32 × 6)-matrix that starts
⎛
⎞
1 1 1 1 1 1
1 1 1 1 1 −1
1 1 1 −1 1 1
1 1 1 −1 1 −1
1 −1 1 1 1 1
C =
1 −1 1 1 1 −1
1 −1 1 −1 1 1
,
1 −1 1 −1 1 −1
−1 −1 1 1 1 1
⎜ −1 −1 1 1 1 −1
⎟
⎝
. . . . . .
⎠
x
H 1
H 5
H 6
H 2
where the first eight rows of C are derived from the first row of B, the
second eight rows come from the second row of B, etc.
The point now is that all the rows of C are different: If two rows are derived
from the same row of B, then they are different since their zeros have been
replaced differently; if they are derived from different rows of B, then they
differ no matter how the zeros have been replaced. But the rows of C are
(±1)-vectors of length s, and there are only 2 s different such vectors. Thus
since the rows of C are distinct, C can have at most 2 s rows, that is,
H 3
H 4
The first row of the C-matrix represents
the shaded triangle, while the second
row corresponds to an empty intersection
of the halfspaces. The point x leads
to the vector
`
1 −1 1 1 −1 1
´
that does not appear in the C-matrix.
2 s−d−1 r ≤ 2 s .
However, not all possible (±1)-vectors appear in C, which yields a strict
inequality 2 s−d−1 r < 2 s , and thus r < 2 d+1 . To see this, we note that
every row of C represents an intersection of halfspaces — just as for the
rows of B before, via the formula (∗). This intersection is a subset of the
simplex P i , which was given by the corresponding row of B. Let us take
a point x ∈ R d that does not lie on any of the hyperplanes H j , and not in
any of the simplices P i . From this x we derive a (±1)-vector that records
for each j whether x ∈ H + j or x ∈ H − j . This (±1)-vector does not occur
in C, because its halfspace intersection according to (∗) contains x and thus
is not contained in any simplex P i .
□
References
[1] F. BAGEMIHL: A conjecture concerning neighboring tetrahedra, Amer. Math.
Monthly 63 (1956) 328-329.
[2] V. J. D. BASTON: Some Properties of Polyhedra in Euclidean Space, Pergamon
Press, Oxford 1965.
[3] M. A. PERLES: At most 2 d+1 neighborly simplices in E d , Annals of Discrete
Math. 20 (1984), 253-254.
[4] J. ZAKS: Neighborly families of 2 d d-simplices in E d , Geometriae Dedicata
11 (1981), 279-296.
[5] J. ZAKS: No Nine Neighborly Tetrahedra Exist, Memoirs Amer. Math. Soc.
No. 447, Vol. 91, 1991.