v2009.01.01 - Convex Optimization

394 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX
Because (926) and (927) translate,
R n ⊇ A −α = aff(X − α1 T ) = aff(P − α) ⊇ P − α ⊇ {x l − α} (931)
where from the previous relations it is easily shown
aff(P − α) = aff(P) − α (932)
Translating A neither changes its dimension or the dimension of the
embedded polyhedron P ; (69)
r ∆ = dim A = dim(A − α) ∆ = dim(P − α) = dim P (933)
For any α ∈ R n , (929)-(933) remain true. [266, 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 (931), and whose dimension is
more 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 translate
their affine hull A and their convex hull P and, in particular, x 1 ∈ P ⊆ A to
the 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 (934)
where V N is defined in (804), and e 1 in (814). Applying (931) to (934),
R n ⊇ R(XV N ) = A−x 1 = aff(X −x 1 1 T ) = aff(P −x 1 ) ⊇ P −x 1 ∋ 0
(935)
where XV N ∈ R n×N−1 . Hence
r = dim R(XV N ) (936)
Since shifting the geometric center to the origin (5.5.1.0.1) translates the
affine hull to the origin as well, then it must also be true
r = dim R(XV ) (937)