In mathematics, especially convex analysis, the recession cone of a set is a cone containing all vectors such that recedes in that direction. That is, the set extends outward in all the directions given by the recession cone.
Mathematical definition Given a nonempty set for some vector space , then the recession cone is given by
If is additionally a convex set then the recession cone can equivalently be defined by
If is a nonempty closed convex set then the recession cone can equivalently be defined as for any choice of
Properties If is a nonempty set then . If is a nonempty convex set then is a convex cone. If is a nonempty closed convex subset of a finite-dimensional Hausdorff space (e.g. ), then if and only if is bounded. If is a nonempty set then where the sum denotes Minkowski addition.
Relation to asymptotic cone The asymptotic cone for is defined by
By the definition it can easily be shown that
In a finite-dimensional space, then it can be shown that if is nonempty, closed and convex. In infinite-dimensional spaces, then the relation between asymptotic cones and recession cones is more complicated, with properties for their equivalence summarized in.
Sum of closed sets Dieudonné's theorem: Let nonempty closed convex sets a locally convex space, if either or is locally compact and is a linear subspace, then is closed. Let nonempty closed convex sets such that for any then , then is closed.
See also Barrier cone
References