Dual cone and polar cone
where <y, x> is the duality pairing between X and X*, i.e. <y, x> = y(x).
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.
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.
- C1 ⊆ C2 implies .
- 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)
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 than 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.
For a set C in X, the polar cone of C is the set
It can be seen that the polar cone is equal to the negative of the dual cone, i.e. Co = −C*.
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