v2007.11.26 - Convex Optimization

54 CHAPTER 2. CONVEX GEOMETRYIn particular, we define affine dimension r of the N-point list X asdimension of the smallest affine set in Euclidean space R n that contains X ;r ∆ = dim aff X (67)Affine dimension r is a lower bound sometimes called embedding dimension.[268] [134] That affine set A in which those points are embedded is uniqueand called the affine hull [46,2.1.2] [249,2.1];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 (68)parallel to subspacewhereR{x l − x 1 , l=2... N} = R(X − x 1 1 T ) ⊆ R n (69)R(A) = {Ax | ∀x} (121)Given some arbitrary set C and any x∈ Cwhere aff(C−x) is a subspace.aff C = x + aff(C − x) (70)2.3.1.0.1 Definition. Affine subset.We analogize affine subset to subspace, 2.14 defining it to be any nonemptyaffine set (2.1.4).△The affine hull of a point x is that point itself;aff ∅ ∆ = ∅ (71)aff{x} = {x} (72)The affine hull of two distinct points is the unique line through them.(Figure 15) The affine hull of three noncollinear points in any dimensionis that unique plane containing the points, and so on. The subspace ofsymmetric matrices S m is the affine hull of the cone of positive semidefinitematrices; (2.9)aff S m + = S m (73)2.14 The popular term affine subspace is an oxymoron.