v2007.09.13 - Convex Optimization

2.1. CONVEX SET 392.1.7.1 Line intersection with boundaryA line can intersect the boundary of a convex set in any dimension at apoint demarcating the line’s entry to the set interior. On one side of thatentry-point along the line is the exterior of the set, on the other side is theset interior. In other words,starting from any point of a convex set, a move toward the interior isan immediate entry into the interior. [20,II.2]When a line intersects the interior of a convex body in any dimension, theboundary appears to the line to be as thin as a point. This is intuitivelyplausible because, for example, a line intersects the boundary of the ellipsoidsin Figure 10 at a point in R , R 2 , and R 3 . Such thinness is a remarkablefact when pondering visualization of convex polyhedra (2.12,5.14.3) infour dimensions, for example, having boundaries constructed from otherthree-dimensional convex polyhedra called faces.We formally define face in (2.6). For now, we observe the boundaryof a convex body to be entirely constituted by all its faces of dimensionlower than the body itself. For example: The ellipsoids in Figure 10 haveboundaries composed only of zero-dimensional faces. The two-dimensionalslab in Figure 9 is a polyhedron having one-dimensional faces making itsboundary. The three-dimensional bounded polyhedron in Figure 12 haszero-, one-, and two-dimensional polygonal faces constituting its boundary.2.1.7.1.1 Example. Intersection of line with boundary in R 6 .The convex cone of positive semidefinite matrices S 3 + (2.9) in the ambientsubspace of symmetric matrices S 3 (2.2.2.0.1) is a six-dimensional Euclideanbody in isometrically isomorphic R 6 (2.2.1). The boundary of thepositive semidefinite cone in this dimension comprises faces having only thedimensions 0, 1, and 3 ; id est, {ρ(ρ+1)/2, ρ=0, 1, 2}.Unique minimum-distance projection PX (E.9) of any point X ∈ S 3 onthat cone is known in closed form (7.1.2). Given, for example, λ ∈ int R 3 +and diagonalization (A.5.2) of exterior point⎡X = QΛQ T ∈ S 3 , Λ =∆ ⎣⎤λ 1 0λ 2⎦ (16)0 −λ 3