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:

C₁ ⊆ C₂ implies .