Dual cone and polar cone are closely related concepts in convex analysis, a branch of mathematics.
The dual cone
of a subset
in a Euclidean space
is the set
where "·" denotes the dot product.
is always a convex cone, even if is neither convex nor a cone.
When is a cone, the following properties hold:
- A non-zero vector is in if and only if is the normal of a hyperplane that supports at the origin.
- is closed and convex.
- implies .
- If has nonempty interior, then is pointed, i.e. contains no line in its entirety.
- If the closure of is pointed, then has nonempty interior.
- is the closure of the smallest convex cone containing .
A cone is said to be self-dual if . The nonnegative orthant of and the space of all positive semidefinite matrices are self-dual.
Dual cones can be more generally defined on real Hilbert spaces.
For a set in the polar cone of is the set
It is easy to check that for any set in and that the polar cone shares many of the properties of the dual cone.
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- Boltyanski, V. G.; Martini, H., Soltan, P. Excursions into combinatorial geometry. New York: Springer.
- Ramm, A.G.; Shivakumar, P.N.; Strauss, A.V. editors Operator theory and its applications. Providence, R.I.: American Mathematical Society.