v2009.01.01 - Convex Optimization

2.3. HULLS 59
2.3.2 Convex hull
The convex hull [173,A.1.4] [266] of any bounded 2.15 list or set of N points
X ∈ R n×N forms a unique bounded convex polyhedron (confer2.12.0.0.1)
whose vertices constitute some subset of that list;
P ∆ = conv {x l , l=1... N} = conv X = {Xa | a T 1 = 1, a ≽ 0} ⊆ R n
(78)
Union of relative interior and relative boundary (2.6.1.3) of the polyhedron
comprise its convex hull P , the smallest closed convex set that contains the
list X ; e.g., Figure 16. Given P , the generating list {x l } is not unique.
But because every bounded polyhedron is the convex hull of its vertices,
[286,2.12.2] the vertices of P comprise a minimal set of generators.
Given some arbitrary set C ⊆ R n , its convex hull conv C is equivalent to
the smallest closed convex set containing it. (confer2.4.1.1.1) The convex
hull is a subset of the affine hull;
conv C ⊆ aff C = aff C = aff C = aff conv C (79)
Any closed bounded convex set C is equal to the convex hull of its boundary;
C = conv ∂ C (80)
conv ∅ ∆ = ∅ (81)
2.3.2.0.1 Example. Hull of outer product. [119] [249] [10,4.1]
[253,3] [206,2.4] [207] Convex hull of the set comprising outer product of
orthonormal 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 +
(82)
This important convex body we call Fantope (after mathematician Ky Fan).
In case k = 1, there is slight simplification: ((1504), Example 2.9.2.4.1)
conv { UU T | U ∈ R N , U T U = 1 } = { A∈ S N | A ≽ 0, 〈I , A〉=1 } (83)
2.15 A set in R n is bounded iff it can be contained in a Euclidean ball having finite radius.
[96,2.2] (confer5.7.3.0.1)