v2009.01.01 - Convex Optimization

5.7. EMBEDDING IN AFFINE HULL 393
5.7 Embedding in affine hull
The affine hull A (70) of a point list {x l } (arranged columnar in X ∈ R n×N
(68)) is identical to the affine hull of that polyhedron P (78) formed from all
convex combinations of the x l ; [53,2] [266,17]
A = aff X = aff P (926)
Comparing hull definitions (70) and (78), it becomes obvious that the x l
and their convex hull P are embedded in their unique affine hull A ;
A ⊇ P ⊇ {x l } (927)
Recall: affine dimension r is a lower bound on embedding, equal to
dimension of the subspace parallel to that nonempty affine set A in which
the points are embedded. (2.3.1) We define dimension of the convex hull P
to be the same as dimension r of the affine hull A [266,2], but r is not
necessarily equal to rank of X (946).
For the particular example illustrated in Figure 92, P is the triangle in
union with its relative interior while its three vertices constitute the entire
list X . The affine hull A is the unique plane that contains the triangle, so
r = 2 in that example while the rank of X is 3. Were there only two points
in Figure 92, then the affine hull would instead be the unique line passing
through them; r would become 1 while rank would then be 2.
5.7.1 Determining affine dimension
Knowledge of affine dimension r becomes important because we lose any
absolute offset common to all the generating x l in R n when reconstructing
convex polyhedra given only distance information. (5.5.1) To calculate r , we
first remove any offset that serves to increase dimensionality of the subspace
required to contain polyhedron P ; subtracting any α ∈ A in the affine hull
from every list member will work,
translating A to the origin: 5.30
X − α1 T (928)
A − α = aff(X − α1 T ) = aff(X) − α (929)
P − α = conv(X − α1 T ) = conv(X) − α (930)
5.30 The manipulation of hull functions aff and conv follows from their definitions.