v2007.09.13 - Convex Optimization

70 CHAPTER 2. CONVEX GEOMETRYnonempty interior) 2.17 at each point on its boundary.2.4.2.6.1 Definition. Supporting hyperplane ∂H .The partial boundary ∂H of a closed halfspace that contains arbitrary set Yis called a supporting hyperplane ∂H to Y when the hyperplane contains atleast one point of Y . [228,11] Specifically, given normal a≠0 (belongingto H + by definition), the supporting hyperplane to Y at y p ∈ ∂Y [sic] is∂H − = { y | a T (y − y p ) = 0, y p ∈ Y , a T (z − y p ) ≤ 0 ∀z∈Y }= { y | a T y = sup{a T z |z∈Y} } (108)where normal a and set Y reside in opposite halfspaces. (Figure 20(a)) Realfunctionσ Y (a) ∆ = sup{a T z |z∈Y} (458)is called the support function for Y .An equivalent but nontraditional representation 2.18 for a supportinghyperplane is obtained by reversing polarity of normal a ; (1454)∂H + = { y | ã T (y − y p ) = 0, y p ∈ Y , ã T (z − y p ) ≥ 0 ∀z∈Y }= { y | ã T y = − inf{ã T z |z∈Y} = sup{−ã T z |z∈Y} } (109)where normal ã and set Y now both reside in H + . (Figure 20(b))When a supporting hyperplane contains only a single point of Y , thathyperplane is termed strictly supporting (and termed tangent to Y if thesupporting hyperplane is unique there [228,18, p.169]).△A closed convex set C ⊂ R n , for example, can be expressed as theintersection of all halfspaces partially bounded by hyperplanes supportingit; videlicet, [181, p.135]C = ⋂a∈R n {y | a T y ≤ σ C (a) } (110)by the halfspaces theorem (2.4.1.1.1).2.17 It is conventional to speak of a hyperplane supporting set C but not containing C ;called nontrivial support. [228, p.100] Hyperplanes in support of lower-dimensional bodiesare admitted.2.18 useful for constructing the dual cone; e.g., Figure 42(b). Tradition recognizes thepolar cone; which is the negative of the dual cone.