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.
Given a nonempty set for some vector space X, 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 
- For any nonempty set then .
- For any nonempty convex set then is a convex cone.
- For any nonempty closed convex set where is a finite-dimensional Hausdorff space (e.g. ), then if and only if is bounded.
- For any nonempty set then where the sum is given by 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.
- Rockafellar, R. Tyrrell (1997) . Convex Analysis. Princeton, NJ: Princeton University Press. pp. 60–76. ISBN 978-0-691-01586-6.
- Borwein, Jonathan; Lewis, Adrian (2006). Convex Analysis and Nonlinear Optimization: Theory and Examples (2 ed.). Springer. ISBN 978-0-387-29570-1.
- Zălinescu, Constantin (2002). Convex analysis in general vector spaces (J). River Edge, NJ: World Scientific Publishing Co., Inc. pp. 6–7. ISBN 981-238-067-1. MR 1921556.
- Kim C. Border. "Sums of sets, etc." (pdf). Retrieved March 7, 2012.
- Alfred Auslender; M. Teboulle (2003). Asymptotic cones and functions in optimization and variational inequalities. Springer. pp. 25–80. ISBN 978-0-387-95520-9.
- Zălinescu, Constantin (1993). "Recession cones and asymptotically compact sets". Journal of Optimization Theory and Applications (Springer Netherlands) 77 (1): 209–220. doi:10.1007/bf00940787. ISSN 0022-3239.
- J. Dieudonné (1966). "Sur la séparation des ensembles convexes". Math. Ann. 163.