v2009.01.01 - Convex Optimization

166 CHAPTER 2. CONVEX GEOMETRY
Γ 4
Γ 2
Γ 3
K ∗
x
z
K y
Γ 1
0
Γ 1 ⊥ Γ 4
Γ 2 ⊥ Γ 3
K ∗
w − K
u
w
Figure 55: (confer Figure 139) Simplicial cone K in R 2 and its dual K ∗ drawn
truncated. Conically independent generators Γ 1 and Γ 2 constitute extreme
directions of K while Γ 3 and Γ 4 constitute extreme directions of K ∗ . Dotted
ray-pairs bound translated cones K . Point x is comparable to point z
(and vice versa) but not to y ; z ≽ K
x ⇔ z − x ∈ K ⇔ z − x ≽ K
0 iff ∃
nonnegative coordinates for biorthogonal expansion of z − x . Point y is not
comparable to z because z does not belong to y ± K . Translating a negated
cone is quite helpful for visualization: u ≼ K
w ⇔ u ∈ w − K ⇔ u − w ≼ K
0.
Points need not belong to K to be comparable; e.g., all points less than w
(with respect to K) belong to w − K .