Gruber P. Convex and Discrete Geometry

350 Convex Polytopes
of the vectors in B. Letu be such a vector with cu(> 0) as close as possible to 0.
Then, in particular, u �∈ B and we may choose vectors v,w ∈ C ∩ Z d \{o}, with
u = v + w. Then cu = c v + c w. Since v,w ∈ C \{o} we thus have that 0 <
c v,c w < cu which contradicts our choice of u. Thus every vector in C ∩ Z d ,
including o, is a non-negative integer linear combination of vectors in B. The proof
of (2) is complete. ⊓⊔
The notion of geometric Hilbert basis and the geometric Hilbert basis theorem
remind one of the Hilbert [500] basis theorem for ideals in polynomial rings over
Noetherian rings. There are actually relations between these two topics, see [915],
p. 376 for references.
Characterization of Totally dual Integral Systems by Hilbert Bases
We prove the following result, where a row ai of A is active on F if ai x = βi for
each x ∈ F:
Theorem 20.5. Let Ax ≤ b be a rational system of linear inequalities. Then the
following propositions are equivalent:
(i) Ax ≤ b is totally dual integral.
(ii) For each face F of the convex polyhedron P ={x : Ax ≤ b} the rows of A which
are active on F form a Hilbert basis of the cone generated by these rows.
Proof. (i)⇒(ii) Let F be a face of P and let a1,...,ak be the rows of A which are
active on F. Then, clearly, the following hold:
(3) Let x be a relatively interior point of F. Leta be a row of A and β the
corresponding entry of b. Then
ax= β if a is one of a1,...,ak and ax