v2009.01.01 - Convex Optimization

408 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX
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Figure 107: Elliptope E 2 in isometrically isomorphic R 3 is a line segment
illustrated interior to positive semidefinite cone S 2 + (Figure 37).
5.9.1 Geometric arguments
5.9.1.0.1 Definition. Elliptope: [205] [202,2.3] [96,31.5]
a unique bounded immutable convex Euclidean body in S n ; intersection of
positive semidefinite cone S n + with that set of n hyperplanes defined by unity
main diagonal;
E n ∆ = S n + ∩ {Φ∈ S n | δ(Φ)=1} (981)
a.k.a, the set of all correlation matrices of dimension
dim E n = n(n −1)/2 in R n(n+1)/2 (982)
An elliptope E n is not a polyhedron, in general, but has some polyhedral
faces and an infinity of vertices. 5.42 Of those, 2 n−1 vertices (some extreme
points of the elliptope) are extreme directions yy T of the positive semidefinite
cone, where the entries of vector y ∈ R n belong to {±1} and exercise every
combination. Each of the remaining vertices has rank belonging to the set
{k>0 | k(k + 1)/2 ≤ n}. Each and every face of an elliptope is exposed.
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5.42 Laurent defines vertex distinctly from the sense herein (2.6.1.0.1); she defines vertex
as a point with full-dimensional (nonempty interior) normal cone (E.10.3.2.1). Her
definition excludes point C in Figure 27, for example.