Dual cone and polar cone

Dual cone and polar cone are closely related concepts in convex analysis, a branch of mathematics. Dual cone and polar cone are closely related concepts in convex analysis, a branch of mathematics. The dual cone C* of a subset C in a linear space X, e.g. Euclidean space Rn, with topological dual space X* is the set where ⟨ y , x ⟩ {displaystyle langle y,x angle } is the duality pairing between X and X*, i.e. ⟨ y , x ⟩ = y ( x ) {displaystyle langle y,x angle =y(x)} . C* is always a convex cone, even if C is neither convex nor a 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. Using this latter definition for C*, we have that when C is a cone, the following properties hold: 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