Dual cone and polar cone
Dual cone and polar cone are closely related concepts in convex analysis, a branch of mathematics.
Dual cone
The dual cone C* of a subset C in a linear space X, e.g. Euclidean space Rn, with topological dual space X* is the set
where <y, x> is the duality pairing between X and X*, i.e. <y, x> = y(x).
C* is always a convex cone, even if C is neither convex nor a cone.
Alternatively, many authors define the dual cone in the context of a real Hilbert space, (such as Rn equipped with the Euclidean inner product) to be what is sometimes called the internal dual cone.
Using this latter definition for C*, we have that when C is a cone, the following properties hold:
A non-zero vector y is in C* if and only if both of the following conditions hold:
y is a normal at the origin of a hyperplane that supports C.
y and C lie on the same side of that supporting hyperplane.
C* is closed and convex.
C1 ⊆ C2 implies 
.
If C has nonempty interior, then C* is pointed, i.e. C* contains no line in its entirety.
