v2010.10.26 - Convex Optimization

72 CHAPTER 2. CONVEX GEOMETRYBy convention, the aberration [330,2.1]cone ∅ {0} (104)Given some arbitrary set C , it is apparentconv C ⊆ cone C (105)2.3.4 Vertex-descriptionThe conditions in (78), (86), and (103) respectively define an affinecombination, convex combination, and conic combination of elements fromthe set or list. Whenever a Euclidean body can be described as somehull or span of a set of points, then that representation is loosely calleda vertex-description and those points are called generators.2.4 Halfspace, HyperplaneA two-dimensional affine subset is called a plane. An (n −1)-dimensionalaffine subset in R n is called a hyperplane. [307] [199] Every hyperplanepartially bounds a halfspace (which is convex, but not affine, and the onlynonempty convex set in R n whose complement is convex and nonempty).2.4.1 Halfspaces H + and H −Euclidean space R n is partitioned in two by any hyperplane ∂H ; id est,H − + H + = R n . The resulting (closed convex) halfspaces, both partiallybounded by ∂H , may be described as an asymmetryH − = {y | a T y ≤ b} = {y | a T (y − y p ) ≤ 0} ⊂ R n (106)H + = {y | a T y ≥ b} = {y | a T (y − y p ) ≥ 0} ⊂ R n (107)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 25) of the hyperplane while acuteangles denote y in H + on the other side.