v2009.01.01 - Convex Optimization

116 CHAPTER 2. CONVEX GEOMETRY
2.9.2.5 Positive semidefinite cone is generally not circular
Extreme angle equation (210) suggests that the positive semidefinite cone
might be invariant to rotation about its axis of revolution; id est, a circular
cone. We investigate this now:
2.9.2.5.1 Definition. Circular cone: 2.38
a pointed closed convex cone having hyperspherical sections orthogonal to
its axis of revolution about which the cone is invariant to rotation. △
A conic section is the intersection of a cone with any hyperplane. In three
dimensions, an intersecting plane perpendicular to a circular cone’s axis of
revolution produces a section bounded by a circle. (Figure 40) A prominent
example of a circular cone in convex analysis is the Lorentz cone (160). We
also find that the positive semidefinite cone and cone of Euclidean distance
matrices are circular cones, but only in low dimension.
The positive semidefinite cone has axis of revolution that is the ray
(base 0) through the identity matrix I . Consider the set of normalized
extreme directions of the positive semidefinite cone: for some arbitrary
positive constant a∈ R +
{yy T ∈ S M | ‖y‖ = √ a} ⊂ ∂S M + (214)
The distance from each extreme direction to the axis of revolution is the
radius
√
R ∆ = inf
c ‖yyT − cI‖ F = a
1 − 1 M
(215)
which is the distance from yy T to a M I ; the length of vector yyT − a M I .
Because distance R (in a particular dimension) from the axis of revolution
to each and every normalized extreme direction is identical, the extreme
directions lie on the boundary of a hypersphere in isometrically isomorphic
R M(M+1)/2 . From Example 2.9.2.4.1, the convex hull (excluding the vertex
at the origin) of the normalized extreme directions is a conic section
C ∆ = conv{yy T | y ∈ R M , y T y = a} = S M + ∩ {A∈ S M | 〈I , A〉 = a} (216)
2.38 A circular cone is assumed convex throughout, although not so by other authors. We
also assume a right circular cone.