v2010.10.26 - Convex Optimization

462 CHAPTER 5. EUCLIDEAN DISTANCE MATRIXIn terms of V N , the unit simplex (293) in R N−1 has an equivalentrepresentation:S = {s ∈ R N−1 | √ 2V N s ≽ −e 1 } (1082)where e 1 is as in (907). Incidental to the EDM assertion, shifting theunit-simplex domain in R N−1 translates the polyhedron P in R n . Indeed,there is a map from vertices of the unit simplex to members of the listgenerating P ;p : R N−1⎛⎧⎪⎨p⎜⎝⎪⎩−βe 1 − βe 2 − β.e N−1 − β⎫⎞⎪⎬⎟⎠⎪⎭→=⎧⎪⎨⎪⎩R nx 1 − αx 2 − αx 3 − α.x N − α⎫⎪⎬⎪⎭(1083)5.9.1.0.5 Proof. EDM assertion.(⇒) We demonstrate that if D is an EDM, then each distance-square ‖p(y)‖ 2described by (1080) corresponds to a point p in some embedded polyhedronP − α . Assume D is indeed an EDM; id est, D can be made from some listX of N unknown points in Euclidean space R n ; D = D(X) for X ∈ R n×Nas in (891). Since D is translation invariant (5.5.1), we may shift the affinehull A of those unknown points to the origin as in (1023). Then take anypoint p in their convex hull (86);P − α = {p = (X − Xb1 T )a | a T 1 = 1, a ≽ 0} (1084)where α = Xb ∈ A ⇔ b T 1 = 1. Solutions to a T 1 = 1 are: 5.47a ∈{e 1 + √ }2V N s | s ∈ R N−1(1085)where e 1 is as in (907). Similarly, b = e 1 + √ 2V N β .P − α = {p = X(I − (e 1 + √ 2V N β)1 T )(e 1 + √ 2V N s) | √ 2V N s ≽ −e 1 }= {p = X √ 2V N (s − β) | √ 2V N s ≽ −e 1 } (1086)5.47 Since R(V N )= N(1 T ) and N(1 T )⊥ R(1) , then over all s∈ R N−1 , V N s is ahyperplane through the origin orthogonal to 1. Thus the solutions {a} constitute ahyperplane orthogonal to the vector 1, and offset from the origin in R N by any particularsolution; in this case, a = e 1 .