v2007.09.13 - Convex Optimization

5.7. EMBEDDING IN AFFINE HULL 3355.7 Embedding in affine hullThe affine hull A (67) of a point list {x l } (arranged columnar in X ∈ R n×N(65)) is identical to the affine hull of that polyhedron P (75) formed from allconvex combinations of the x l ; [46,2] [228,17]A = aff X = aff P (826)Comparing hull definitions (67) and (75), it becomes obvious that the x land their convex hull P are embedded in their unique affine hull A ;A ⊇ P ⊇ {x l } (827)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 hullP to be the same as dimension r of the affine hull A [228,2], but r is notnecessarily equal to the rank of X (846).For the particular example illustrated in Figure 73, P is the triangle plusits relative interior while its three vertices constitute the entire list X . Theaffine hull A is the unique plane that contains the triangle, so r=2 in thatexample while the rank of X is 3. Were there only two points in Figure 73,then the affine hull would instead be the unique line passing through them;r would become 1 while the 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: 5.29X − α1 T (828)A − α = aff(X − α1 T ) = aff(X) − α (829)P − α = conv(X − α1 T ) = conv(X) − α (830)5.29 The manipulation of hull functions aff and conv follows from their definitions.