v2010.10.26 - Convex Optimization

42 CHAPTER 2. CONVEX GEOMETRY2.1.7.1 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. [26,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 13 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 Euclidean dimensions, for example, having boundaries constructed fromother three-dimensional convex polyhedra called faces.We formally define face in (2.6). For now, we observe the boundary ofa convex body to be entirely constituted by all its faces of dimension lowerthan the body itself. Any face of a convex set is convex. For example: Theellipsoids in Figure 13 have boundaries composed only of zero-dimensionalfaces. The two-dimensional slab in Figure 11 is an unbounded polyhedronhaving one-dimensional faces making its boundary. The three-dimensionalbounded polyhedron in Figure 20 has zero-, one-, and two-dimensionalpolygonal 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 S 3 + is known in closed form (7.1.2). Given, for example, λ∈int R 3 +and diagonalization (A.5.1) of exterior point⎡ ⎤λ 1 0X = QΛQ T ∈ S 3 , Λ ⎣ λ 2⎦ (21)0 −λ 3