v2010.10.26 - Convex Optimization

2.3. HULLS 61Figure 20: Convex hull of a random list of points in R 3 . Some pointsfrom that generating list reside interior to this convex polyhedron (2.12).[373, Convex Polyhedron] (Avis-Fukuda-Mizukoshi)2.3 HullsWe focus on the affine, convex, and conic hulls: convex sets that may beregarded as kinds of Euclidean container or vessel united with its interior.2.3.1 Affine hull, affine dimensionAffine dimension of any set in R n is the dimension of the smallest affine set(empty set, point, line, plane, hyperplane (2.4.2), translated subspace, R n )that contains it. For nonempty sets, affine dimension is the same as dimensionof the subspace parallel to that affine set. [307,1] [199,A.2.1]Ascribe the points in a list {x l ∈ R n , l=1... N} to the columns ofmatrix X :X = [x 1 · · · x N ] ∈ R n×N (76)In 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 (77)Affine dimension r is a lower bound sometimes called embedding dimension.[352] [185] That affine set A in which those points are embedded is uniqueand called the affine hull [330,2.1];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 (78)