v2009.01.01 - Convex Optimization

56 CHAPTER 2. CONVEX GEOMETRY
Ascribe the points in a list {x l ∈ R n , l=1... N} to the columns of
matrix X :
X = [x 1 · · · x N ] ∈ R n×N (68)
In particular, we define affine dimension r of the N-point list X as
dimension of the smallest affine set in Euclidean space R n that contains X ;
r ∆ = dim aff X (69)
Affine dimension r is a lower bound sometimes called embedding dimension.
[305] [159] That affine set A in which those points are embedded is unique
and called the affine hull [53,2.1.2] [286,2.1];
A ∆ = aff {x l ∈ R n , l=1... N} = aff X
= x 1 + R{x l − x 1 , l=2... N} = {Xa | a T 1 = 1} ⊆ R n (70)
parallel to subspace
where
R{x l − x 1 , l=2... N} = R(X − x 1 1 T ) ⊆ R n (71)
R(A) = {Ax | ∀x} (132)
Given some arbitrary set C and any x∈ C
where aff(C−x) is a subspace.
aff C = x + aff(C − x) (72)
2.3.1.0.1 Definition. Affine subset.
We analogize affine subset to subspace, 2.14 defining it to be any nonempty
affine set (2.1.4).
△
aff ∅ = ∆ ∅ (73)
The affine hull of a point x is that point itself;
aff{x} = {x} (74)
2.14 The popular term affine subspace is an oxymoron.