Gruber P. Convex and Discrete Geometry

20 Linear Optimization 347
of E k . They generate a lattice L in S. We show that b ′ ∈ S\L. Letx ∈ F. Then
b ′ = A ′ x = x1b ′ 1 +···+xdb ′ d ∈ S. Ifb′ ∈ L, then b ′ = u1b ′ 1 +···+udb ′ d = A′ u
for suitable u ∈ Z d . Hence F contains the integer vector u, which is excluded by
assumption. Let L ∗ be the polar lattice of L in S. Since b ′ �∈ L there is a rational row
y ′ ∈ L ∗ with
y ′ b ′ �∈ Z, while y ′ A ′ = (y ′ b ′ 1 ,...,y′ b ′ d ) ∈ Zd
by the definition of L ∗ . By adding suitable positive integers to the entries of y ′ ,if
necessary, we may suppose that
If a1,...,ak are the rows of A ′ , then
y ′ ≥ o while still y ′ b ′ �∈ Z, y ′ A ′ ∈ Z d .
c = y ′ A ′ = y ′ 1 a1 +···+y ′ k ak ∈ pos{a1,...,ak}∩Z d = NP(F) ∩ Z d ,
β = y ′ b ′ ∈ Q \ Z
by Proposition 20.1. The hyperplane
H ={x : cx = β} ={x : y ′ A ′ x = y ′ b ′ }
is rational, contains F ={x : A ′ x = b ′ } and its normal vector c is in NP(F). Thus
H supports P. H contains no u ∈ Z d since otherwise y ′ b ′ = y ′ A ′ u ∈ Z while
y ′ b ′ �∈ Z. This contradicts (ii) and thus concludes the proof. ⊓⊔
A consequence of this result is the following:
Corollary 20.1. Let Ax ≤ b be a rational system of linear inequalities. Then the
following statements are equivalent:
(i) sup{cx : Ax ≤ b} is attained by an integer vector x for each rational row c for
which the supremum is finite.
(ii) sup{cx : Ax ≤ b} is an integer for each integer row c for which the supremum
is finite.
(iii) P ={x : Ax ≤ b} is a lattice polyhedron.
Proof. (i)⇒(ii) Let c be an integer row and such that sup{cx : Ax ≤ b} is finite. By
(i), the supremum is attained at an integer vector x and thus is an integer.
(ii)⇒(iii) We first show the following:
(2) Let H ={x : cx = δ} be a rational support hyperplane of P. Then H
contains a point of Z d .
By multiplying the equation cx = δ by a suitable positive rational number and
changing notation, if necessary, we may suppose that c is an integer row vector with
relatively prime entries. Since H is a support hyperplane of P, sup{cx : Ax ≤ b} =
δ is finite. By (ii), δ is then an integer. Since c has relatively prime integer entries,
there is a u ∈ Z d such that cu = δ. Hence u ∈ H, concluding the proof of (2).
Having proved (2), Theorem 20.3 implies statement (iii).
(iii)⇒(i) Trivial. ⊓⊔