Gruber P. Convex and Discrete Geometry

14 Preliminaries and the Face Lattice 249
If (i) or (ii) hold, then C is a closed convex cone with apex o. Hence Proposition 3.3
shows that
C = (C ∩ L ⊥ ) ⊕ L,
where L is the linearity space of C and C ∩ L ⊥ is a pointed closed convex cone with
apex o. We may assume that L ={x : x1 = ··· = xc = 0}, L ⊥ ={x : xc+1 =
···= xd = 0}. Then it is easy to see that
C is a convex
An application of (1) then implies (2).
�
V
�
-cone ⇐⇒ C ∩ L
H
⊥ is a convex
(3) Let P ⊆ E d . Then the following are equivalent:
(i) P is a convex V-polyhedron.
(ii) P is a convex H-polyhedron.
�
V
�
-cone.
H
Embed E d into E d+1 as usual and let u = (o, 1) ∈ E d+1 . Consider P + u ⊆ E d+1
and let C be the smallest closed convex cone with apex o containing P + u.
(i)⇒(ii) Let P = conv{x1,...,xm}+pos{y1,...,yn}. Then
C = � (λ1x1 +···+λmxm,λ)+ µ1y1 +···+µn yn :
λi ≥ 0, λ1 +···+λm = λ, µj ≥ 0 �
is a convex V- and thus a convex H-cone by (2). Then
P + u = C ∩ � (x, z) : z ≥ 1, −z ≥−1 �
is a convex H-polyhedron. This, in turn, shows that P is a convex H-polyhedron.
(ii)⇒(i) Let P ={x : Ax ≤ b}. Then
C = � (x, z) : z ≥ 0, Ax − bz ≤ o �
is an H-cone and thus a V-cone by (2), say
C = pos � �
(x1, 1), . . . , (xm, 1), y1,...,yn
with suitable xi, yi ∈ E d . Then
and thus
P + u = C ∩ (E d + u)
= � (λ1x1 +···+λmxm, 1) + µ1y1 +···+µn yn :
λi ≥ 0, λ1 +···+λm = 1, µj ≥ 0 �
P = � λ1x1 +···+λmxm + µ1y1 +···+µn yn :
λi ≥ 0, λ1 +···+λm = 1, µj ≥ 0 � .
The proof of (3) and thus of the theorem is complete. ⊓⊔