Chapter 1
Our proof of the Main Theorem starts by assuming, on the contrary, that there exists a
neighborly family F in
E3,
consisting of nine tetrahedra P1? P2, ... , P9. Let {H1? ..., Ht}
denote the collection of all the planes in E3 which contain facets of members of F. Let H+j and
H"j denote the two closed half-spaces, determined by Hj, for all j .
Following Baston [2] (see also [17, 12, 20, 21, 23]), let B(F) = (by) be the Baston matrix
of F, defined by
/ +1 if H. contains a facet of P and P c
H+
,
I J i 1 j
b = - 1 if H. contains a facet of P and P. c H7 ,
ij I J i 1 J
V 0 otherwise,
for all l i 9 and 1 jt.
The following properties hold:
(1) Each row of B(F) has precisely four nonzero terms.
(2) For every pair of row-indices i and j , there exists a unique column-index k, such that
bik-bjk = - 1 -
Property (1) follows from the fact that every tetrahedron has four facets, while property (2)
is implied by the neighborliness of F, see [2,17].
4
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