Gruber P. Convex and Discrete Geometry

346 Convex Polytopes
Rational Polyhedra and Lattice Polyhedra
A convex polyhedron P in E d is rational, if it has a representation of the form
P ={x : Ax ≤ b},
where A is a rational matrix and b a rational vector. The convex polyhedron P is
called integer, integral or a lattice polyhedron if it is the convex hull of the points of
the integer lattice Z d contained in it. It is not difficult to show the following equivalences,
where a minimum face of a convex polyhedron is a face which does not
contain a proper subface. Thus a minimum face is a vertex, a line or a plane of
dimension ≥ 2.
Proposition 20.3. Let P be a rational convex polyhedron in E d . Then the following
statements are equivalent:
(i) P is a lattice polyhedron.
(ii) Each face of P contains a point of Z d .
(iii) Each minimum face of P contains a point of Z d .
(iv) If in a linear optimization problem of the form sup{cx : x ∈ P} the supremum is
finite, it is attained at a point of Z d .
(v) Each support hyperplane of P contains a point of Z d .
Some Properties of Lattice Polyhedra
The following result of Edmonds and Giles [287] contains a series of earlier results
as special cases.
Theorem 20.3. Let P be a rational convex polyhedron in E d . Then the following
statements are equivalent:
(i) P is a lattice polyhedron.
(ii) Each rational support hyperplane of P contains a point of Z d .
Proof. (i)⇒(ii) Clear.
(ii)⇒(i) We may assume that P ={x : Ax ≤ b}, where A, b are integer. It is
sufficient to show that each minimum face of P contains a point of Z d . To see this,
assume that, on the contrary, there is a minimum face F of P which contains no point
of Z d . Being a minimum face, F is a plane and thus can be represented in the form
F ={x : A ′ x = b ′ },
where the matrix A ′ consists of, say k (≤ m) rows of A and the column b ′ of the corresponding
entries of b. We now construct a rational hyperplane H which supports
P but contains no point of Z d in contradiction to (ii). For this we need the definition
and simple properties of polar lattices in Sect. 21.4. The columns b ′ 1 ,...,b′ d of
A ′ are integer vectors in E k and are contained in the subspace S = lin{b ′ 1 ,...,b′ d }