Dual cone and polar cone

Pure MathematicsQ2687694

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

∗

=

{

y
∈

X

∗

:
⟨
y
,
x
⟩
≥
0

∀
x
∈
C

}

,

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

where

⟨
y
,
x
⟩

{\displaystyle \langle y,x\rangle }

is the duality pairing between X and X*, i.e.

⟨
y
,
x
⟩
=
y
(
x
)

{\displaystyle \langle y,x\rangle =y(x)}

.
The set

C

∗

{\displaystyle C^{*}}

is always a convex cone, even if

C

{\displaystyle 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
∈

X

′

:
Re
⁡

(

f
(
x
)

)

≥
0

for all

x
∈
C

}

{\displaystyle 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

′

{\displaystyle C^{\prime }}

will be a convex cone.
If C ⊆ {0} then

C

′

=

X

′

{\displaystyle 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

internal

∗

:=

{

y
∈
X
:
⟨
y
,
x
⟩
≥
0

∀
x
∈
C

}

.

{\displaystyle 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:
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

⊆

C

2

{\displaystyle C_{1}\subseteq C_{2}}

implies

C

2

∗

⊆

C

1

∗

{\displaystyle 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

=

{

y
∈

X

∗

:
⟨
y
,
x
⟩
≤
0

∀
x
∈
C

}

.

{\displaystyle 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.

Dual cone and polar cone

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