Dual cone and polar cone



Dual cone and polar cone are closely related concepts in convex analysis, a branch of mathematics.

In a vector space
The dual cone C$$ of a subset C in a linear space X over the reals, e.g. Euclidean space Rn, with dual space X$$ is the set


 * $$C^* = \left \{y\in X^*: \langle y, x \rangle \geq 0 \quad \forall x\in C \right \},$$

where $$\langle y, x \rangle$$ is the duality pairing between X and X$$, i.e. $$\langle y, x\rangle = y(x)$$.

C$$ is always a convex cone, even if C is neither convex nor a cone.

In a topological vector space
If X is a topological vector space over the real or complex numbers, then the dual cone of a subset C ⊆ X is the following set of continuous linear functionals on X:


 * $$C^{\prime} := \left\{ f \in X^{\prime} : \operatorname{Re} \left( f (x) \right) \geq 0 \text{ for all } x \in C \right\}$$,

which is the polar of the set -C. No matter what C is, $$C^{\prime}$$ will be a convex cone. If C ⊆ {0} then $$C^{\prime} = X^{\prime}$$.

In a Hilbert space (internal dual 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.


 * $$C^*_\text{internal} := \left \{y\in X: \langle y, x \rangle \geq 0 \quad \forall x\in C \right \}.$$

Properties
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:
 * 1) y is a normal at the origin of a hyperplane that supports C.
 * 2) y and C lie on the same side of that supporting hyperplane.
 * C$$ is closed and convex.
 * $$C_1 \subseteq C_2$$ implies $$C_2^* \subseteq C_1^*$$.
 * If C has nonempty interior, then C$$ is pointed, i.e. C* contains no line in its entirety.
 * If C is a cone and the closure of C is pointed, then C$$ has nonempty interior.
 * C$$ is the closure of the smallest convex cone containing C (a consequence of the hyperplane separation theorem)

Self-dual cones
A cone C in a vector space X is said to be self-dual if X can be equipped with an inner product ⟨⋅,⋅⟩ such that the internal dual cone relative to this inner product is equal to C. Those authors who define the dual cone as the internal dual cone in a real Hilbert space usually say that a cone is self-dual if it is equal to its internal dual. This is slightly different from the above definition, which permits a change of inner product. For instance, the above definition makes a cone in Rn with ellipsoidal base self-dual, because the inner product can be changed to make the base spherical, and a cone with spherical base in Rn is equal to its internal dual.

The nonnegative orthant of Rn and the space of all positive semidefinite matrices are self-dual, as are the cones with ellipsoidal base (often called "spherical cones", "Lorentz cones", or sometimes "ice-cream cones"). So are all cones in R3 whose base is the convex hull of a regular polygon with an odd number of vertices. A less regular example is the cone in R3 whose base is the "house": the convex hull of a square and a point outside the square forming an equilateral triangle (of the appropriate height) with one of the sides of the square.

Polar cone
For a set C in X, the polar cone of C is the set


 * $$C^o = \left \{y\in X^*: \langle y, x \rangle \leq 0 \quad \forall x\in C \right \}.$$

It can be seen that the polar cone is equal to the negative of the dual cone, i.e. Co = −C$$.

For a closed convex cone C in X, the polar cone is equivalent to the polar set for C.