v2009.01.01 - Convex Optimization

62 CHAPTER 2. CONVEX GEOMETRY
‖xy T ‖=1
‖uv T ‖=1
xy T
uv T
‖uv T ‖≤1
{X ∈ R m×n | ∑ i
σ(X) i ≤ 1}
0
−uv T
‖−uv T ‖=1
−xy T
‖−xy T ‖=1
Figure 19: uv T is a convex combination of normalized dyads ‖±uv T ‖=1 ;
similarly for xy T . Any point in line segment joining xy T to uv T is expressible
as a convex combination of two to four points indicated on boundary.
(⇒) Now suppose we are given a convex combination of dyads
X = ∑ ∑
α i u i vi
T such that αi =1, α i ≥ 0 ∀i, and ‖u i vi T ‖≤1 ∀i.
∑
Then by triangle inequality for singular values [177, cor.3.4.3]
σ(X)i ≤ ∑ σ(α i u i vi T )= ∑ α i ‖u i vi T ‖≤ ∑ α i .
Given any particular dyad uv T in the convex hull, because its polar
−uv T and every convex combination of the two belong to that hull, then the
unique line containing those two points (their affine combination (70)) must
intersect the hull’s boundary at the normalized dyads {±uv T | ‖uv T ‖=1}.
Any point formed by convex combination of dyads in the hull must therefore
be expressible as a convex combination of dyads on the boundary: e.g.,
Figure 19.
conv{uv T | ‖uv T ‖ ≤ 1, u∈ R m , v ∈ R n } ≡ conv{uv T | ‖uv T ‖ = 1, u∈ R m , v ∈ R n }
(87)
id est, dyads may be normalized and the hull’s boundary contains them;
∂{X ∈ R m×n | ∑ i
σ(X) i ≤ 1} ⊇ {uv T | ‖uv T ‖ = 1, u∈ R m , v ∈ R n } (88)