Convex Cone - Dual Cone

Let be a convex cone in a real vector space V equipped with an inner product. The dual cone to C is the set This is also a convex cone. If C is equal to its dual cone, C is called self-dual.

Another common notion of the dual of a cone is that it is the cone in 's dual space defined by:

In other words, if is the algebraic dual space of, it is the set of linear functionals that are nonnegative on the primal cone . If we take to be the continuous dual space then it is the set of continuous linear functionals nonnegative on . This notion does not require the specification of an inner product on .

In finite dimensions, the two notions of dual cone are essentially the same because any inner product induces a linear isomorphism (nonsingular linear map) from to, and this isomorphism will take the dual cone given by the second definition, in, onto the one given by the first definition. A cone can be said to be self-dual without reference to any given inner product, if there exists an inner product with respect to which it is equal to its dual by the first definition. The map from to induced by this inner product will therefore take to . However, the existence of an linear isomorphism of the dual cone onto the primal cone is not equivalent to self-duality in this sense: while every such isomorphism induces a nonsingular bilinear form on, this form is not necessarily positive definite (i.e., not necessarily an inner product). There are many examples of cones that are linearly isomorphic to their dual cones, but not self-dual: any cone in three-dimensions with a regular polygonal base having an even number of vertices is an example.