v2007.09.13 - Convex Optimization

336 CHAPTER 5. EUCLIDEAN DISTANCE MATRIXBecause (826) and (827) translate,R n ⊇ A −α = aff(X − α1 T ) = aff(P − α) ⊇ P − α ⊇ {x l − α} (831)where from the previous relations it is easily shownaff(P − α) = aff(P) − α (832)Translating A neither changes its dimension or the dimension of theembedded polyhedron P ; (66)r ∆ = dim A = dim(A − α) ∆ = dim(P − α) = dim P (833)For any α ∈ R n , (829)-(833) remain true. [228, p.4, p.12] Yet when α ∈ A ,the affine set A − α becomes a unique subspace of R n in which the {x l − α}and their convex hull P − α are embedded (831), and whose dimension ismore easily calculated.5.7.1.0.1 Example. Translating first list-member to origin.Subtracting the first member α = ∆ x 1 from every list member will translatetheir affine hull A and their convex hull P and, in particular, x 1 ∈ P ⊆ A tothe origin in R n ; videlicet,X − x 1 1 T = X − Xe 1 1 T = X(I − e 1 1 T ) = X[0 √ ]2V N ∈ R n×N (834)where V N is defined in (711), and e 1 in (721). Applying (831) to (834),R n ⊇ R(XV N ) = A−x 1 = aff(X −x 1 1 T ) = aff(P −x 1 ) ⊇ P −x 1 ∋ 0(835)where XV N ∈ R n×N−1 . Hencer = dim R(XV N ) (836)Since shifting the geometric center to the origin (5.5.1.0.1) translates theaffine hull to the origin as well, then it must also be truer = dim R(XV ) (837)