Английская Википедия:Dual cone and 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 CШаблон:Sup of a subset C in a linear space X over the reals, e.g. Euclidean space Rn, with dual space XШаблон:Sup is the set
- <math>C^* = \left \{y\in X^*: \langle y , x \rangle \geq 0 \quad \forall x\in C \right \},</math>
where <math>\langle y, x \rangle</math> is the duality pairing between X and XШаблон:Sup, i.e. <math>\langle y, x\rangle = y(x)</math>.
CШаблон: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:
- <math>C^{\prime} := \left\{ f \in X^{\prime} : \operatorname{Re} \left( f (x) \right) \geq 0 \text{ for all } x \in C \right\}</math>,Шаблон:Sfn
which is the polar of the set -C.Шаблон:Sfn No matter what C is, <math>C^{\prime}</math> will be a convex cone. If C ⊆ {0} then <math>C^{\prime} = X^{\prime}</math>.
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.
- <math>C^*_\text{internal} := \left \{y\in X: \langle y , x \rangle \geq 0 \quad \forall x\in C \right \}.</math>
Properties
Using this latter definition for CШаблон:Sup, we have that when C is a cone, the following properties hold:[1]
- A non-zero vector y is in CШаблон:Sup 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.
- CШаблон:Sup is closed and convex.
- <math>C_1 \subseteq C_2</math> implies <math>C_2^* \subseteq C_1^*</math>.
- If C has nonempty interior, then CШаблон: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 CШаблон:Sup has nonempty interior.
- CШаблон: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.[2] 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[3]
- <math>C^o = \left \{y\in X^*: \langle y , x \rangle \leq 0 \quad \forall x\in C \right \}.</math>
It can be seen that the polar cone is equal to the negative of the dual cone, i.e. Co = −CШаблон:Sup.
For a closed convex cone C in X, the polar cone is equivalent to the polar set for C.[4]
See also
References
Bibliography
- Шаблон:Cite book
- Шаблон:Cite book
- Шаблон:Narici Beckenstein Topological Vector Spaces
- Шаблон:Cite book
- Шаблон:Schaefer Wolff Topological Vector Spaces
Шаблон:Ordered topological vector spaces
- ↑ Шаблон:Cite book
- ↑ Iochum, Bruno, "Cônes autopolaires et algèbres de Jordan", Springer, 1984.
- ↑ Шаблон:Cite book
- ↑ Шаблон:Cite book
