2. Preliminaries
A subset C of a finite-dimensional real inner product space H is called a
convex cone if it is closed under addition and closed under multiplication by
non-negative scalars; C is a closed convex cone if it is closed in the topology
induced by the inner product. If C is a cone, then
(2.1) C* = {y G H : [x,y] 0 for all x G C}
is the dual cone to C. It is easy to show that C is also a cone, and that
C C (C ) . If C is a closed convex cone and z ^ C, then by Minkowski's Theorem
on separating hyperplanes, there exists y G C so that [z,y] 0. It follows
that C = (C ) if C is a closed convex cone. (A general reference on convexity
is [R9]; for convexity in moment problems, see [K2]and [K10].)
A simple geometric criterion for determining the interior of a closed
convex cone in H uses the compactness of {x G H : ||x||= 1}.
Lemma 2.2
Suppose C C H is a closed convex cone and x G C. Then x G int C ifand
only if 0 # y G C implies [x,y] 0.
If 0 # y G C and [x,y] = 0, then [x - 77y,y] = -7y[y,y] 0, so x - rjy t C
for all TJ 0 and x g int C. If [x,y] 0 for all 0 # y G C , then [x,u] 8
0 on the compact set {u G C : ||u|| = 1}. It follows by linearity that [x,y]
5||y|| for all y G C . Thus, if z G H, ||x - z|| 6 and y G C , then [z,y] =
[x,y] - [x - z,y] % | | - ||x - z||.||y|| 0, so z G (C*)* = C. n
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