Gruber P. Convex and Discrete Geometry

250 Convex Polytopes
Generalized Convex Polyhedra
For the geometric theory of positive definite quadratic forms, see Sect. 29.4, for
Dirichlet-Voronoĭ tilings, see Sect. 32.1, and in other contexts, a more general
notion of convex polyhedron is needed: call a closed convex set in E d a generalized
convex polyhedron if its intersection with any convex polytope is a convex polytope.
A generalized convex polyhedron looks locally like a convex polyhedron, but may
have countably many different faces. An example of a generalized convex polygon
is the set
conv � (u,v)∈ Z 2 : v ≥ u 2� ⊆ E 2 .
Normal Cones of Polytopes and Polyhedra
Let P be a convex polyhedron and p a boundary point of P.Thenormal cone NP(p)
of P at p is the closed convex cone of all exterior normal vectors of support hyperplanes
of P at p, that is,
NP(p) = � u : u · x ≤ u · p for all x ∈ P � .
For later reference we need the following result, where the ai are the row vectors of
the n × d matrix A, theβi the components of the vector b ∈ E n , and the inequality
Ax ≤ b is to be understood componentwise. We consider the row vectors ai as
vectors in E d and thus write ai · u instead of a T i · u for the inner product. Using the
matrix product and considering ai as a row vector, we also write the inner product in
the form aiu.
Proposition 14.1. Let P ={x : Ax ≤ b} be a convex polyhedron and p ∈ bd P.
Assume that ai p = βi precisely for i = 1,...,k. Then
(4) NP(p) = pos � �
a1,...,ak .
If p is a vertex of P, then
(5) dim NP(p) = d.
Proof. Since NP(p) depends on P only locally and is translation invariant, we may
assume that p = o. Then
NP(p) = NC(o), where C is the closed convex cone {x : Bx ≤ o}
and B is the k × d matrix consisting of the rows a1,...,ak. Since each of the hyperplanes
{x : ai · x = ai x = 0} supports C at o, it follows that NC(o) ⊇{a1,...,ak}
and thus
NC(o) ⊇ pos � �
a1,...,ak .
To show that equality holds, assume that, on the contrary, there is c ∈ NC(o) \
pos{a1,...,ak}. Choose a hyperplane {x : u · x = α} which strictly separates c and
the closed convex cone pos{a1,...,ak}, sayu · c >α>u · x for each x of the