Dual cone
From Wikipedia, the free encyclopedia
In convex analysis, the dual cone 
of a set 
, where 
is a Hilbert space, is the set
is always a convex cone, even if 
is neither convex nor a cone. The set is usually a cone in 
, in which case 
if and only if 
is the normal of a hyperplane that supports at the origin. When is a cone, the following properties hold:
- is closed and convex.
implies 
.- If has nonempty interior, then is pointed, i.e. contains no line.
- If the closure of is pointed then has nonempty interior.
is the closure of the convex hull of .
A cone is said to be self-dual if 
. The nonnegative orthant of and the space of all positive semidefinite matrices are self-dual.
