For any set , its anti-cone , with respect to , is the set
The anti-cone is also called the dual cone.
The anti-cone operation is generally applied to subsets of that are themselves cones. Recall that a cone in a real vector space generalize the notion of linear inequalities in a finite number of real variables. The dual cone provides a natural way to transfer such inequalities in the original vector space to its dual space. The concept is useful in the theory of duality.
The anti-cone is a convex cone in .
If is non-negative, then so is for . And if , then clearly for . ∎
If is a cone, then its anti-cone may be equivalently characterized as:
Vectors will be identified with their images under the natural embedding of in its double dual space.
The pairing is sometimes called a dual pair; and , where is identified with its image in the double dual, is also a dual pair.
is weak-* closed.
Properties involving cone inclusion
Property 5 (Farkas’ lemma).
Let be a weakly-closed convex cone. Then if and only if for all .
That for and is just the definition. For the converse, we show that if , then there exists such that .
for any convex cone . (The anti-cone operation on is to be taken with respect to .)
We work with , which is a weakly-closed convex cone. By Property 5, if and only if for all . But by definition of the second anti-cone, if and only if for all . ∎
Let and be convex cones in , with weakly closed. Then if and only if .
|Date of creation||2013-03-22 17:20:48|
|Last modified on||2013-03-22 17:20:48|
|Last modified by||stevecheng (10074)|