Gruber P. Convex and Discrete Geometry

416 Geometry of Numbers
with suitable λ ji ∈ R. Multiplying this representation of ˆb j by bi for i = 1,..., j − 1,
gives the following system of linear equations for λ j1,...,λjj−1:
b j · bi = λ j1b1 · bi +···+λ jj−1b j−1 · bi, i = 1,..., j − 1.
(Note that ˆb j · bi = 0.) This system has determinant D j > 0, see (9). Since
b1,...,bd ∈ L ⊆ Z d and thus all inner products bi · bk are integer, it follows
that D jλji ∈ Z for i = 1,..., j − 1. Since, by (9), the integer D j > 0 divides the
integer D > 0, (12) implies (11). Next we show that
(13) max{� ˆb1�,...,� ˆbd�} ≤ max{�a1�,...,�ad�} = A0,
say. Step I does not change this maximum. In Step II, ĉi = ˆbi for i �= j, j + 1.
Further �ĉ j � < � ˆb j �, by the condition of Step II, see also (6), and �ĉ j+1� ≤�ˆb j �
since ĉ j+1 is a suitable orthogonal projection of ˆb j. Hence Step II never increases the
maximum. This concludes the proof of (13). Since D ≤ D0, (11) and (13) together
yield (10).
Third, we prove the following statement:
(14) For every basis {b1,...,bd} resulting from Steps I or II in the algorithm
every coordinates of any vector b j is an integer bounded by √ dA0.
For the proof it is sufficient to show that:
(15) max{�b1� 2 ,...,�bd� 2 }≤dA 2 0 .
After Step I has been executed, |µ ji|≤ 1 2 . Then (1), together with (13), implies that:
�b j� 2 =
j�
µ 2 ij� ˆbi� 2 ≤
i=1
j�
� ˆbi� 2 ≤ dA 2 0 .
Hence (15) holds after an execution of Step I. Since Step II does not change the
maximum, (15) holds generally, concluding the proof of (14).
The fourth step is to show that:
(16) For every basis {b1,...,bd} resulting from a substep of Step I in the algorithm,
every coordinate of any b j is an integer bounded by (2D0 A0) d√ dA0.
Note that the Gram–Schmidt orthogonalization does not change during an execution
of Step I. Thus, for the bases appearing during Step I, statement (10) is valid. If, in a
substep of Step I, the basis vector b j is replaced by b j −⌈µ ji⌋bi, then
i=1
|µ ji|= |b j · ˆbi|
� ˆbi� 2 ≤ �b j�
� ˆbi� ≤ D0�b j �
by the Cauchy–Schwarz inequality and (10). Thus
|⌈µ ji⌋| ≤ 2|µ ji|≤2D0�b j �