Dual cone and polar cone

A set C and its dual cone C^*.

A set C and its polar cone C^o. The dual cone and the polar cone are symmetric to each other with respect to the origin.

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

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

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

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.

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

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

The polar of the closed convex cone C is the closed convex cone C^o, and vice versa.

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. C^o = −C^*.

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

See also

• Bipolar theorem
• Polar set

Bibliography

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