v2007.11.26 - Convex Optimization

62 CHAPTER 2. CONVEX GEOMETRY2.4.1 Halfspaces H + and H −Euclidean space R n is partitioned into two halfspaces by any hyperplane∂H ; id est, H − + H + = R n . The resulting (closed convex) halfspaces, bothpartially bounded by ∂H , may be describedH − = {y | a T y ≤ b} = {y | a T (y − y p ) ≤ 0} ⊂ R n (87)H + = {y | a T y ≥ b} = {y | a T (y − y p ) ≥ 0} ⊂ R n (88)where nonzero vector a∈R n is an outward-normal to the hyperplane partiallybounding H − while an inward-normal with respect to H + . For any vectory −y p that makes an obtuse angle with normal a , vector y will lie in thehalfspace H − on one side (shaded in Figure 18) of the hyperplane while acuteangles denote y in H + on the other side.An equivalent more intuitive representation of a halfspace comes aboutwhen we consider all the points in R n closer to point d than to point c orequidistant, in the Euclidean sense; from Figure 18,H − = {y | ‖y − d‖ ≤ ‖y − c‖} (89)This representation, in terms of proximity, is resolved with the moreconventional representation of a halfspace (87) by squaring both sides ofthe inequality in (89);}H − ={y | (c − d) T y ≤ ‖c‖2 − ‖d‖ 2=2( {y | (c − d) T y − c + d ) }≤ 02(90)2.4.1.1 PRINCIPLE 1: Halfspace-description of convex setsThe most fundamental principle in convex geometry follows from thegeometric Hahn-Banach theorem [183,5.12] [16,1] [88,I.1.2] whichguarantees any closed convex set to be an intersection of halfspaces.2.4.1.1.1 Theorem. Halfspaces. [46,2.3.1] [232,18][148,A.4.2(b)] [30,2.4] A closed convex set in R n is equivalent to theintersection of all halfspaces that contain it.⋄