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## 4.2.7 Separating hyperplane

A standard result in convex analysis known as the Separating Hyperplane Theorem (see, e.g., [Ber99, Proposition B.13] or [BV04, Section 2.5.1]) says that if and are two nonempty disjoint convex sets then there exists a hyperplane that separates them; by this we mean that is contained in one of the two closed half-spaces created by the hyperplane and is contained in the other. The ray is a convex set, and from the convexity of the terminal cone it is easy to see that its interior is convex as well. Lemma 4.1 guarantees that does not intersect the interior of . Therefore, we can apply the Separating Hyperplane Theorem to conclude the existence of a hyperplane separating from the interior of , and hence from itself.4.3Obviously, this separating hyperplane must pass through the point which is a common point of and . Such a hyperplane need not be unique. The normal to the hyperplane is a nonzero vector in (it is defined up to a constant multiple once we fix the hyperplane). Let us denote this normal vector by (4.28)

where and are, by definition, its -component and -component, respectively. Then the equation of the hyperplane is and the separation property is formally written as such that (4.29)

and (4.30)

where is the vector (4.26) which generates the ray . Note that if we were to flip the direction of the normal vector (4.28), the inequality signs in (4.29) and (4.30) would be reversed; the present choice is simply a matter of sign convention (cf. Section 3.4.4). For an illustration, see Figure 4.11 in which the shaded object represents the separating hyperplane. In view of the definition (4.26) of , the inequality (4.30) simply says that , as required by the statement of the maximum principle. This will be the only use of (4.30). The normal vector (4.28) will serve as the terminal condition for the adjoint system, to be defined next.     Next: 4.2.8 Adjoint equation Up: 4.2 Proof of the Previous: 4.2.6 Key topological lemma   Contents   Index
Daniel 2010-12-20