Gruber P. Convex and Discrete Geometry

It is sufficient to show that
(5) {e} =H1 ∩···∩ Hk.
14 Preliminaries and the Face Lattice 247
If this did not hold, the flat H = H1 ∩···∩Hk has dimension at least 1 and we could
choose u,v ∈ H, int H −
k+1 ,...,int H − m , u,v �= e, and such that e = 1 2 (u + v). Then
u,v ∈ P and we obtain a contradiction to the assumption that e is an extreme point
of P. This proves (5) and thus concludes the proof of (4). ⊓⊔
Corollary 14.1. Let P, Q ∈ P. Then P ∩ Q ∈ P.
14.2 Extension to Convex Polyhedra and Birkhoff’s Theorem
Many combinatorial and geometric results on convex polytopes have natural extensions
to convex polyhedra.
In the following we generalize some of the definitions and the main result of the
last section to convex polyhedra. As a tool, which will be needed later, we give a
simple representation of normal cones of polyhedra. An application of the latter is a
short geometric proof of Birkhoff’s theorem 5.7 on doubly stochastic matrices.
V-Polyhedra and H-Polyhedra
AsetP in E d is a convex V-polyhedron if there are finite sets {x1,...,xm} and
{y1,...,yn} in E d such that
where
P = conv{x1,...,xm}+pos{y1,...,yn}
= � λ1x1 +···+λmxm : λi ≥ 0, λ1 +···+λm = 1 �
+ � µ1y1 +···+µn yn : µ j ≥ 0 �
= Q + � {µR : µ ≥ 0} =Q + C,
Q = � λ1x1 +···+λmxm : λi ≥ 0, λ1 +···+λm = 1 � ,
R = � µ1y1 +···+µn yn : µ j ≥ 0, µ1 +···+µn = 1 � ,
C = � µ1y1 +···+µn yn : µ j ≥ 0 � .
Thus P is the sum of a convex polytope Q and a closed convex cone C with apex o
and therefore a closed convex set.
AsetP in E d is a convex H-polyhedron if it is the intersection of finitely many
closed halfspaces, that is
P ={x ∈ E d : Ax ≤ b},
where A is a real m×d matrix, b ∈ E m , and the inequality is to be understood componentwise.
This definition abundantly meets the requirements of linear optimization.
A bounded convex H-polyhedron is a convex H-polytope.