v2010.10.26 - Convex Optimization

68 CHAPTER 2. CONVEX GEOMETRY‖xy T ‖=1xy T p‖uv T ‖=1uv T p‖uv T ‖≤1{X ∈ R m×n | ∑ iσ(X) i ≤ 1}0−uv T p‖−uv T ‖=1−xy T p‖−xy T ‖=1Figure 23: uv T p is a convex combination of normalized dyads ‖±uv T ‖=1 ;similarly for xy T p . Any point in line segment joining xy T p to uv T p is expressibleas a convex combination of two to four points indicated on boundary. Affinedimension of vectorized nuclear norm ball in X∈R 2×2 is 2 because (1610)σ 1 = ‖u‖‖v‖=1 by (95).