v2010.10.26 - Convex Optimization

102 CHAPTER 2. CONVEX GEOMETRY2.7.2.1 cone invarianceMore Euclidean bodies are cones, it seems, than are not. 2.28 This class ofconvex body, the convex cone, is invariant to scaling, linear and single- ormany-valued inverse linear transformation, vector summation, and Cartesianproduct, but is not invariant to translation. [307, p.22]2.7.2.1.1 Theorem. Cone intersection (nonempty).Intersection of an arbitrary collection of convex cones is a convex cone.[307,2,19]Intersection of an arbitrary collection of closed convex cones is a closedconvex cone. [258,2.3]Intersection of a finite number of polyhedral cones (Figure 50 p.143,2.12.1.0.1) remains a polyhedral cone.⋄The property pointedness is associated with a convex cone; but,pointed cone convex cone (Figure 35, Figure 36)2.7.2.1.2 Definition. Pointed convex cone. (confer2.12.2.2)A convex cone K is pointed iff it contains no line. Equivalently, K is notpointed iff there exists any nonzero direction Γ ∈ K such that −Γ ∈ K . Ifthe origin is an extreme point of K or, equivalently, ifK ∩ −K = {0} (179)then K is pointed, and vice versa. [330,2.10] A convex cone is pointed iffthe origin is the smallest nonempty face of its closure.△Then a pointed closed convex cone, by principle of separating hyperplane(2.4.2.7), has a strictly supporting hyperplane at the origin. The simplestand only bounded [371, p.75] convex cone K = {0} ⊆ R n is pointed, byconvention, but not full-dimensional. Its relative boundary is the empty set∅ (25) while its relative interior is the point 0 itself (12). The pointed convex2.28 confer Figures: 24 33 34 35 38 37 39 41 43 50 54 57 59 60 62 63 64 65 66 138151 175