NEIGHBORLY TETRAHEDRA
5
Following [17, 20, 21], let xy, i j, denote the number of columns of B(F), consisting of i
nonzero terms of one sign and j nonzero terms of the opposite sign, besides zeros. If x0j 0, for
j 1, we cut j-1 of the tetrahedra, having a facet on a common plane, by parallel planes, such that
the neighborliness is preserved and such that the j facets lie on different planes. We may assume,
therefore, that XQJ 0 implies that j = 1. The quantity XQJ is just the total number offree facets
(for definition, see [17]) of members of F.
The following are known:
(3) Xjj 0 implies j 3.
(4) x3)3 = 0.
Property (3) is Lemma 16 in [2, p.56], while property (4) is one of the main steps in [2],
proved as Theorem 7 in p. 157, there.
A member of F is said to be of type (p, q, r, s), p q r s, if it touches (in a
two-dimensional set) p other members of F in one facet, q of them in another facets, etc. Thus,
there are just four possible types: (3, 3, 2, 0), (3, 3, 1, 1), (3, 2, 2, 1) and (2, 2, 2, 2). Let a,
(3, y and 8 denote the number of members of F of types (3, 3, 2, 0), (3, 3, 1, 1), (3, 2, 2, 1)
and (2, 2, 2, 2), respectively. Using Baston's nomenclature, a is the number of "naiks", p - of
"dhoats", y - of "thaiks" and 8 - of "chardhos".
Clearly, only the members of F of type (3, 3, 2, 0) have free facets, one each.
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