v2007.09.13 - Convex Optimization

2.4. HALFSPACE, HYPERPLANE 59of dimension k(N −√k(1 k + 1) and radius − k ) centered at kI along2 2 N Nthe ray (base 0) through the identity matrix I in isomorphic vector spaceR N(N+1)/2 (2.2.2.1).Figure 14 illustrates extreme points (81) comprising the boundary of aFantope, the boundary of a disc corresponding to k = 1, N = 2 ; but thatcircumscription does not hold in higher dimension. (2.9.2.5) 2.3.3 Conic hullIn terms of a finite-length point list (or set) arranged columnar in X ∈ R n×N(65), its conic hull is expressedK ∆ = cone {x l , l=1... N} = coneX = {Xa | a ≽ 0} ⊆ R n (83)id est, every nonnegative combination of points from the list. The conic hullof any finite-length list forms a polyhedral cone [147,A.4.3] (2.12.1.0.1;e.g., Figure 15); the smallest closed convex cone that contains the list.By convention, the aberration [245,2.1]Given some arbitrary set C , it is apparent2.3.4 Vertex-descriptioncone ∅ ∆ = {0} (84)conv C ⊆ cone C (85)The conditions in (67), (75), and (83) 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.2.4 Halfspace, HyperplaneA two-dimensional affine subset is called a plane. An (n −1)-dimensionalaffine subset in R n is called a hyperplane. [228] [147] Every hyperplanepartially bounds a halfspace (which is convex but not affine).