v2009.01.01 - Convex Optimization

66 CHAPTER 2. CONVEX GEOMETRY
2.4 Halfspace, Hyperplane
A two-dimensional affine subset is called a plane. An (n −1)-dimensional
affine subset in R n is called a hyperplane. [266] [173] Every hyperplane
partially bounds a halfspace (which is convex but not affine).
2.4.1 Halfspaces H + and H −
Euclidean space R n is partitioned into two halfspaces by any hyperplane ∂H ;
id est, H − + H + = R n . The resulting (closed convex) halfspaces, both
partially bounded by ∂H , may be described
H − = {y | a T y ≤ b} = {y | a T (y − y p ) ≤ 0} ⊂ R n (97)
H + = {y | a T y ≥ b} = {y | a T (y − y p ) ≥ 0} ⊂ R n (98)
where nonzero vector a∈R n is an outward-normal to the hyperplane partially
bounding H − while an inward-normal with respect to H + . For any vector
y −y p that makes an obtuse angle with normal a , vector y will lie in the
halfspace H − on one side (shaded in Figure 21) of the hyperplane while acute
angles denote y in H + on the other side.
An equivalent more intuitive representation of a halfspace comes about
when we consider all the points in R n closer to point d than to point c or
equidistant, in the Euclidean sense; from Figure 21,
H − = {y | ‖y − d‖ ≤ ‖y − c‖} (99)
This representation, in terms of proximity, is resolved with the more
conventional representation of a halfspace (97) by squaring both sides of
the inequality in (99);
}
(
H − =
{y | (c − d) T y ≤ ‖c‖2 − ‖d‖ 2
=
{y | (c − d) T y − c + d ) }
≤ 0
2
2
(100)
2.4.1.1 PRINCIPLE 1: Halfspace-description of convex sets
The most fundamental principle in convex geometry follows from the
geometric Hahn-Banach theorem [215,5.12] [17,1] [110,I.1.2] which
guarantees any closed convex set to be an intersection of halfspaces.