v2007.09.13 - Convex Optimization

2.3. HULLS 57The union of relative interior and relative boundary (2.6.1.3) of thepolyhedron comprise the convex hull P , the smallest closed convex set thatcontains the list X ; e.g., Figure 12. Given P , the generating list {x l } isnot unique.Given some arbitrary set C ⊆ R n , its convex hull conv C is equivalent tothe smallest closed convex set containing it. (confer2.4.1.1.1) The convexhull is a subset of the affine hull;conv C ⊆ aff C = aff C = aff C = aff conv C (76)Any closed bounded convex set C is equal to the convex hull of its boundary;C = conv ∂ C (77)conv ∅ ∆ = ∅ (78)2.3.2.0.1 Example. Hull of outer product. [211] [9,4.1][215,3] [174,2.4] Convex hull of the set comprising outer product oforthonormal matrices has equivalent expression: for 1 ≤ k ≤ N (2.9.0.1)conv { UU T | U ∈ R N×k , U T U = I } = { A∈ S N | I ≽ A ≽ 0, 〈I , A〉=k } ⊂ S N +(79)This important convex body we call Fantope (after mathematician Ky Fan).In case k = 1, there is slight simplification: ((1398), Example 2.9.2.4.1)conv { UU T | U ∈ R N , U T U = I } = { A∈ S N | A ≽ 0, 〈I , A〉=1 } (80)In case k = N , the Fantope is identity matrix I . More generally, the set{UU T | U ∈ R N×k , U T U = I} (81)comprises the extreme points (2.6.0.0.1) of its convex hull. By (1264), eachand every extreme point UU T has only k nonzero eigenvalues λ and they allequal 1 ; id est, λ(UU T ) 1:k = λ(U T U) = 1. So the Frobenius norm of eachand every extreme point equals the same constant‖UU T ‖ 2 F = k (82)Each extreme point simultaneously lies on the boundary of the positivesemidefinite cone (when k < N ,2.9) and on the boundary of a hypersphere