v2009.01.01 - Convex Optimization

36 CHAPTER 2. CONVEX GEOMETRY
Figure 11: A slab is a convex Euclidean body infinite in extent but
not affine. Illustrated in R 2 , it may be constructed by intersecting
two opposing halfspaces whose bounding hyperplanes are parallel but not
coincident. Because number of halfspaces used in its construction is finite,
slab is a polyhedron. (Cartesian axes and vector inward-normal to each
halfspace-boundary are drawn for reference.)
2.1.4 affine set
A nonempty affine set (from the word affinity) is any subset of R n that is a
translation of some subspace. An affine set is convex and open so contains
no boundary: e.g., empty set ∅ , point, line, plane, hyperplane (2.4.2),
subspace, etcetera. For some parallel 2.4 subspace M and any point x∈A
A is affine ⇔ A = x + M
= {y | y − x∈M}
(9)
The intersection of an arbitrary collection of affine sets remains affine. The
affine hull of a set C ⊆ R n (2.3.1) is the smallest affine set containing it.
2.1.5 dimension
Dimension of an arbitrary set S is the dimension of its affine hull; [324, p.14]
dim S ∆ = dim aff S = dim aff(S − s) , s∈ S (10)
the same as dimension of the subspace parallel to that affine set aff S when
nonempty. Hence dimension (of a set) is synonymous with affine dimension.
[173, A.2.1]
2.4 Two affine sets are said to be parallel when one is a translation of the other. [266, p.4]