Gruber P. Convex and Discrete Geometry

246 Convex Polytopes
V-Polytopes and H-Polytopes
Convex polytopes as defined earlier are also called convex V-polytopes. HereV
stands for vertices. Dually, a convex H-polyhedron is the intersection of finitely many
closed halfspaces. A bounded convex H-polyhedron is called a convex H-polytope.
A formal proof of the following folk theorem is due to Weyl [1021], see also
Minkowski [744], Sect. 4.
Theorem 14.2. Let P ⊆ E d . Then the following statements are equivalent:
(i) P is a convex V-polytope.
(ii) P is a convex H-polytope.
Proof. (i)⇒(ii) We may suppose that dim P = d. P has only finitely many faces.
Since each boundary point of P is contained in a support hyperplane of P by Theorem
4.1, bd P is the union of its proper faces. Connecting an interior point of P with
a line segment which misses all faces of dimension at most d − 2 with an exterior
point, each point where it intersects the boundary of P must be contained in a face
of dimension d − 1, i.e. in a facet. Thus P must have facets. Let Hi, i = 1,...,m,
be the hyperplanes containing the facets of P and H −
i the corresponding support
halfspaces. We claim that
(3) P = H −
1 ∩···∩ H − m .
The inclusion P ⊆ H −
1 ∩ ··· ∩ H − m is trivial. To show the reverse inclusion, let
x ∈ Ed \P. For each of the finitely many faces of P of dimension at most d − 2,
consider the affine hull of the face and x. Choose a point y ∈ int P which is contained
in none of these affine hulls. The intersection of the line segment [x, y] with bd P
then is a point z ∈ bd P which is contained in none of these affine hulls and thus in
none of the faces of P of dimension at most d − 2. Since bd P is the union of all
faces, z is contained in a suitable facet and thus in one of the hyperplanes, say Hi.
Then x �∈ H −
i and therefore x �∈ H −
1 ∩···∩ H − −
m . Hence P ⊇ H1 ∩···∩ H − m ,
concluding the proof of (3).
(ii)⇒(i) Let P = H −
1 ∩···∩ H − −
m be bounded, where each Hi , i = 1, ··· , m,
is a halfspace with boundary hyperplane Hi. Clearly, P is a convex body. By
Minkowski’s theorem 5.5, P is the convex hull of its extreme points. To conclude
the proof that P is a convex polytope, it is thus sufficient to show that P has only
finitely many extreme points. To see this, it is sufficient to prove the following proposition:
(4) Let e be an extreme point of P. Then e is the intersection of a sub-family
of {H1,...,Hm}.
By re-indexing, if necessary, we may assume that
e ∈ H1,...,Hk, int H −
k+1 ,...,int H − m .