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 

Convex analysis