v2010.10.26 - Convex Optimization

5.7. EMBEDDING IN AFFINE HULL 4455.7 Embedding in affine hullThe affine hull A (78) of a point list {x l } (arranged columnar in X ∈ R n×N(76)) is identical to the affine hull of that polyhedron P (86) formed from allconvex combinations of the x l ; [61,2] [307,17]A = aff X = aff P (1021)Comparing hull definitions (78) and (86), it becomes obvious that the x land their convex hull P are embedded in their unique affine hull A ;A ⊇ P ⊇ {x l } (1022)Recall: affine dimension r is a lower bound on embedding, equal todimension of the subspace parallel to that nonempty affine set A in whichthe points are embedded. (2.3.1) We define dimension of the convex hull Pto be the same as dimension r of the affine hull A [307,2], but r is notnecessarily equal to rank of X (1041).For the particular example illustrated in Figure 115, P is the triangle inunion with its relative interior while its three vertices constitute the entirelist X . Affine hull A is the unique plane that contains the triangle, so affinedimension r = 2 in that example while rank of X is 3. Were there only twopoints in Figure 115, then the affine hull would instead be the unique linepassing through them; r would become 1 while rank would then be 2.5.7.1 Determining affine dimensionKnowledge of affine dimension r becomes important because we lose anyabsolute offset common to all the generating x l in R n when reconstructingconvex polyhedra given only distance information. (5.5.1) To calculate r , wefirst remove any offset that serves to increase dimensionality of the subspacerequired to contain polyhedron P ; subtracting any α ∈ A in the affine hullfrom every list member will work,translating A to the origin: 5.33X − α1 T (1023)A − α = aff(X − α1 T ) = aff(X) − α (1024)P − α = conv(X − α1 T ) = conv(X) − α (1025)5.33 The manipulation of hull functions aff and conv follows from their definitions.