v2007.09.13 - Convex Optimization

350 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX0Figure 87: Elliptope E 2 in isometrically isomorphic R 3 is a line segmentillustrated interior to positive semidefinite cone S 2 + (Figure 31).5.9.1 Geometric arguments5.9.1.0.1 Definition. Elliptope: [173] [170,2.3] [77,31.5]a unique bounded immutable convex Euclidean body in S n ; intersection ofpositive semidefinite cone S n + with that set of n hyperplanes defined by unitymain diagonal;E n ∆ = S n + ∩ {Φ∈ S n | δ(Φ)=1} (880)a.k.a, the set of all correlation matrices of dimensiondim E n = n(n −1)/2 in R n(n+1)/2 (881)An elliptope E n is not a polyhedron, in general, but has some polyhedral facesand an infinity of vertices. 5.41 Of those, 2 n−1 vertices are extreme directionsyy T of the positive semidefinite cone where the entries of vector y ∈ R n eachbelong to {±1} while the vector exercises every combination. Each of theremaining vertices has rank belonging to the set {k>0 | k(k + 1)/2 ≤ n}.Each and every face of an elliptope is exposed.△5.41 Laurent defines vertex distinctly from the sense herein (2.6.1.0.1); she defines vertexas a point with full-dimensional (nonempty interior) normal cone (E.10.3.2.1). Herdefinition excludes point C in Figure 21, for example.