v2010.10.26 - Convex Optimization

74 CHAPTER 2. CONVEX GEOMETRYAn 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 25,H − = {y | ‖y − d‖ ≤ ‖y − c‖} (108)This representation, in terms of proximity, is resolved with the moreconventional representation of a halfspace (106) by squaring both sides ofthe inequality in (108);}H − ={y | (c − d) T y ≤ ‖c‖2 − ‖d‖ 2=2( {y | (c − d) T y − c + d ) }≤ 02(109)2.4.1.1 PRINCIPLE 1: Halfspace-description of convex setsThe most fundamental principle in convex geometry follows from thegeometric Hahn-Banach theorem [250,5.12] [18,1] [132,I.1.2] whichguarantees any closed convex set to be an intersection of halfspaces.2.4.1.1.1 Theorem. Halfspaces. [199,A.4.2b] [42,2.4]A closed convex set in R n is equivalent to the intersection of all halfspacesthat contain it.⋄Intersection of multiple halfspaces in R n may be represented using amatrix constant A⋂H i−= {y | A T y ≼ b} = {y | A T (y − y p ) ≼ 0} (110)i⋂H i+= {y | A T y ≽ b} = {y | A T (y − y p ) ≽ 0} (111)iwhere b is now a vector, and the i th column of A is normal to a hyperplane∂H i partially bounding H i . By the halfspaces theorem, intersections likethis can describe interesting convex Euclidean bodies such as polyhedra andcones, giving rise to the term halfspace-description.