Dual cone and polar cone

Dual cone

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^{*} = {y \in X^{*} : ⟨ y, x ⟩ \geq 0 \forall x \in C},

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.

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^{'} := {f \in X^{'} : Re (f (x)) \geq 0 for all x \in C}

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

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_{internal}^{*} := {y \in X : ⟨ y, x ⟩ \geq 0 \forall x \in C} .

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:

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.
$C_{1} \subseteq C_{2}$ 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)

2 more chapters are free with a Lably account.