Polar Cone

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 R^''n'', with dual space ''X'' is the set

:$C^* = \left \,$

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^ := \left\$,

which is the polar of the set -''C''.
No matter what ''C'' is, $C^$ will be a convex cone.
If ''C'' ⊆ 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 R^''n'' equipped with the Euclidean inner product) to be what is sometimes called the ''internal dual cone''.

:$C^*_\text := \left \.$

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)

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 R^''n'' with ellipsoidal base self-dual, because the inner product can be changed to make the base spherical, and a cone with spherical base in R^''n'' is equal to its internal dual.

The nonnegative orthant of R^''n'' 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

:$C^o = \left \.$

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

References

Bibliography

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{{Ordered topological vector spaces

Convex analysis
Convex geometry
Linear programming