Recession cone

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