v2010.10.26 - Convex Optimization

544 CHAPTER 6. CONE OF DISTANCE MATRICESWhen N = 1, the EDM cone and its dual in ambient S h each comprisethe origin in isomorphic R 0 ; thus, selfdual in this dimension. (confer (104))When N = 2, the EDM cone is the nonnegative real line in isomorphic R .(Figure 142) EDM 2∗ in S 2 h is identical, thus selfdual in this dimension.This [ result]is in agreement [ with ](1294), verified directly: for all κ∈ R ,11z = κ and δ(zz−1T ) = κ 2 ⇒ d ∗ 12 ≥ 0.1The first case adverse to selfdualness N = 3 may be deduced fromFigure 138; the EDM cone is a circular cone in isomorphic R 3 correspondingto no rotation of Lorentz cone (178) (the selfdual circular cone). Figure 151illustrates the EDM cone and its dual in ambient S 3 h ; no longer selfdual.6.9 Theorem of the alternativeIn2.13.2.1.1 we showed how alternative systems of generalized inequalitycan be derived from closed convex cones and their duals. This section is,therefore, a fitting postscript to the discussion of the dual EDM cone.6.9.0.0.1 Theorem. EDM alternative. [163,1]Given D ∈ S N hD ∈ EDM Nor in the alternative{1 T z = 1∃ z such thatDz = 0(1297)In words, either N(D) intersects hyperplane {z | 1 T z =1} or D is an EDM;the alternatives are incompatible.⋄When D is an EDM [259,2]N(D) ⊂ N(1 T ) = {z | 1 T z = 0} (1298)Because [163,2] (E.0.1)thenDD † 1 = 11 T D † D = 1 T (1299)R(1) ⊂ R(D) (1300)