v2010.10.26 - Convex Optimization

158 CHAPTER 2. CONVEX GEOMETRYK ∗(a)0K ∗KyK(b)0Figure 55: Two equivalent constructions of dual cone K ∗ in R 2 : (a) Showingconstruction by intersection of halfspaces about 0 (drawn truncated). Onlythose two halfspaces whose bounding hyperplanes have inward-normalcorresponding to an extreme direction of this pointed closed convex coneK ⊂ R 2 need be drawn; by (369). (b) Suggesting construction by union ofinward-normals y to each and every hyperplane ∂H + supporting K . Thisinterpretation is valid when K is convex because existence of a supportinghyperplane is then guaranteed (2.4.2.6).