v2009.01.01 - Convex Optimization

2.3. HULLS 55
Figure 16: Convex hull of a random list of points in R 3 . Some
points from that generating list reside interior to this convex polyhedron.
[326, Convex Polyhedron] (Avis-Fukuda-Mizukoshi)
The set of all symmetric hollow matrices S M h forms a proper subspace
in R M×M , so for it there must be a standard orthonormal basis in
isometrically isomorphic R M(M−1)/2
{E ij ∈ S M h } =
{ }
1 ( )
√ ei e T j + e j e T i , 1 ≤ i < j ≤ M 2
(67)
where M(M −1)/2 standard basis matrices E ij are formed from the standard
basis vectors e i ∈ R M .
The symmetric hollow majorization corollary on page 541 characterizes
eigenvalues of symmetric hollow matrices.
2.3 Hulls
We focus on the affine, convex, and conic hulls: convex sets that may be
regarded as kinds of Euclidean container or vessel united with its interior.
2.3.1 Affine hull, affine dimension
Affine 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 dimension
of the subspace parallel to that affine set. [266,1] [173,A.2.1]