Dual cone and polar cone

A set C and its dual cone CTemplate:Sup.

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 CTemplate:Sup of a subset C in a linear space X over the reals, e.g. Euclidean space Rⁿ, with dual space XTemplate:Sup is the set

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

where $⟨y,x⟩$ is the duality pairing between X and XTemplate:Sup, i.e. $⟨y,x⟩=y(x)$.

CTemplate:Sup 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:

$C^{\prime }:=\{f\in X^{\prime }:\mathrm{Re}(f(x))\ge 0\text{ for all }x\in C\}$,Template:Sfn

which is the polar of the set -C.Template:Sfn 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.

$C_{\text{internal}}^{*}:=\{y\in X:⟨y,x⟩\ge 0\mathrm{\forall }x\in C\}.$

Properties

Using this latter definition for CTemplate:Sup, we have that when C is a cone, the following properties hold:

• A non-zero vector y is in CTemplate:Sup 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.

• CTemplate:Sup is closed and convex.
• $C_1\subseteq C_2$ implies $C_2^{*}\subseteq C_1^{*}$.
• If C has nonempty interior, then CTemplate:Sup is pointed, i.e. C* contains no line in its entirety.
• If C is a cone and the closure of C is pointed, then CTemplate:Sup has nonempty interior.
• CTemplate:Sup 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

$C^o=\{y\in X^{*}:⟨y,x⟩\le 0\mathrm{\forall }x\in C\}.$

It can be seen that the polar cone is equal to the negative of the dual cone, i.e. C^o = −CTemplate:Sup.

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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• Template:Narici Beckenstein Topological Vector Spaces
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• Template:Schaefer Wolff Topological Vector Spaces