v2009.01.01 - Convex Optimization

68 CHAPTER 2. CONVEX GEOMETRY
2.4.1.1.1 Theorem. Halfspaces. [173,A.4.2(b)] [37,2.4]
A closed convex set in R n is equivalent to the intersection of all halfspaces
that contain it.
⋄
Intersection of multiple halfspaces in R n may be represented using a
matrix constant A
⋂
H i−
= {y | A T y ≼ b} = {y | A T (y − y p ) ≼ 0} (101)
i
⋂
H i+
= {y | A T y ≽ b} = {y | A T (y − y p ) ≽ 0} (102)
i
where b is now a vector, and the i th column of A is normal to a hyperplane
∂H i partially bounding H i . By the halfspaces theorem, intersections like
this can describe interesting convex Euclidean bodies such as polyhedra and
cones, giving rise to the term halfspace-description.
2.4.2 Hyperplane ∂H representations
Every hyperplane ∂H is an affine set parallel to an (n −1)-dimensional
subspace of R n ; it is itself a subspace if and only if it contains the origin.
dim∂H = n − 1 (103)
so a hyperplane is a point in R , a line in R 2 , a plane in R 3 , and so on.
Every hyperplane can be described as the intersection of complementary
halfspaces; [266,19]
∂H = H − ∩ H + = {y | a T y ≤ b , a T y ≥ b} = {y | a T y = b} (104)
a halfspace-description. Assuming normal a∈ R n to be nonzero, then any
hyperplane in R n can be described as the solution set to vector equation
a T y = b (illustrated in Figure 21 and Figure 22 for R 2 )
∂H ∆ = {y | a T y = b} = {y | a T (y−y p ) = 0} = {Zξ+y p | ξ ∈ R n−1 } ⊂ R n (105)
All solutions y constituting the hyperplane are offset from the nullspace of
a T by the same constant vector y p ∈ R n that is any particular solution to
a T y=b ; id est,
y = Zξ + y p (106)