Gruber P. Convex and Discrete Geometry

326 Convex Polytopes
V (S) = V .Let j be the largest index such that p1− p0,...,p j−1− p0 ∈ Z d are such
that the parallelotope spanned by these vectors contains no point of Z d except for its
vertices. An application of Theorem 21.3 shows that there is a basis {c1,...,cd} of
Z d such that
pi − p0 = ui1c1 +···+uiici for i = 1,...,d
with suitable uik ∈ Zd . The assumption on j shows that o is the only point of Zd in the parallelotope � α1(p1 − p0) +···+αj−1(pj−1 − p0) : 0 ≤ αi < 1 � and
that |u jj| > 1. The former can be used to show that |u11| = ··· = |u j−1, j−1| =1.
By replacing ci by −ci, if necessary, and renaming, we may assume that u11 =
··· = u j−1, j−1 = 1 and u jj > 1. Putting d1 = c1, d2 = u21c1 + c2,...,d j−1 =
u j−1,1c1 +···+u j−1, j−2c j−2 + c j−1 and d j = u1d1 +···+u jdj−1 + c j with
suitable ui ∈ Zd , and d j+1 = c j+1,...,dd = cd, we obtain a basis � �
d1,...,dd of
Zd such that
p1 − p0 = d1
p2 − p0 = d2
............
p j−1 − p0 = d j−1
p j − p0 = v j1d1 +···+v jjd j
.....................................
pd − p0 = vd1d1 +·········+vdddd
with suitable vik ∈ Z, where 0 ≤ v j1,...,vj, j−1 ≤ v jj,vjj > 1. Finally, permuting
d1,...,d j−1 suitably, if necessary, and retaining d j,...,dd, we obtain a basis
e1,...,ed of Z d such that
V (p1 − p0) = e1
V (p2 − p0) = e2
...............
V (pj−1 − p0) = e j−1
V (pj − p0) = w j1e1 +···+w jje j
..........................................
V (pd − p0) = wd1e1 +·········+wdded
with a suitable integer unimodular d × d matrix V and integers wik ∈ Z such that
0 ≤ w j1 ≤ w j2 ≤ ··· ≤ w jj,wjj > 1. Now choose an integer unimodular d × d
matrix W such that Wei = bi and put U = WV. Then
U(pi − p0) = bi for i = 1,..., j − 1,
U(pj − p0) = w j1b1 +···+w jjb j,
where w jk ∈ Z, 0 ≤ w j1 ≤···≤w jj,wjj > 1.
Since S + Hd p = US− Up0 + Hd p , by the definition of Hdp , we may assume that S
already has this form. The facets of S are
Fi = conv{p0,...,pi−1, pi+1,...,pd}, i = 0,...d.
If p ∈ Z d , represent conv � S ∪{p} � as follows: