Gruber P. Convex and Discrete Geometry

14 Preliminaries and the Face Lattice 251
form x = λ1a1 +···+λkak, λi ≥ 0. Since this holds for all λi ≥ 0, it follows
that u · c > 0 ≥ u · ai = ai · u for i = 1,...,k and thus u ∈ C. Fromu ∈ C and
c ∈ NC(o) we conclude that u · c ≤ 0, a contradiction. Hence, equality holds and the
proof of (4) is complete.
Now suppose that p is a vertex of P. If (5) did not hold, then by (4) we may
choose x �= o such that
while still
ai · (p ± x) = ai · p ± ai · x = ai · p + 0 = βi for i = 1,...,k,
ai · (p ± x) 1 and that it holds for
d − 1.
�d may be interpreted as the subset of Ed2 defined by the following equalities
and inequalities
�
xij = 1, �
xij = 1, xij ≥ 0fori, j = 1,...,d.
i
j
Since this subset is bounded, it is a convex polytope. Consider the affine sub-space S
of Ed2 defined by the 2d hyperplanes
�
xij = 1, �
xij = 1fori, j = 1,...,d.
i
j
To see that dim S = (d − 1) 2 , note that among the coefficient matrices
⎛ ⎞
1, 0,...,0
⎜ 1, 0,...,0 ⎟
⎝ ......... ⎠
1, 0,...,0
,...,
⎛ ⎞
0, 0,...,1
⎜ 0, 0,...,1 ⎟
⎝ ......... ⎠
0, 0,...,1
,
⎛ ⎞
1, 1,...,1
⎜ 0, 0,...,0 ⎟
⎝ ......... ⎠
0, 0,...,0
,...,
⎛ ⎞
0, 0,...,0
⎜ 0, 0,...,0 ⎟
⎝ ......... ⎠
1, 1,...,1
of the 2d hyperplanes there are precisely 2d − 1 linearly independent ones. �d is a
proper convex polytope in the (d−1) 2 -dimensional affine sub-space S, defined by the