Recession cone

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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.[1]

Mathematical definition[edit]

Given a nonempty set for some vector space , then the recession cone is given by

[2]

If is additionally a convex set then the recession cone can equivalently be defined by

[3]

If is a nonempty closed convex set then the recession cone can equivalently be defined as

for any choice of [3]

Properties[edit]

  • If is a nonempty set then .
  • If is a nonempty convex set then is a convex cone.[3]
  • If is a nonempty closed convex subset of a finite-dimensional Hausdorff space (e.g. ), then if and only if is bounded.[1][3]
  • If is a nonempty set then where the sum denotes Minkowski addition.

Relation to asymptotic cone[edit]

The asymptotic cone for is defined by

[4][5]

By the definition it can easily be shown that [4]

In a finite-dimensional space, then it can be shown that if is nonempty, closed and convex.[5] In infinite-dimensional spaces, then the relation between asymptotic cones and recession cones is more complicated, with properties for their equivalence summarized in.[6]

Sum of closed sets[edit]

  • 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.[7][3]
  • Let nonempty closed convex sets such that for any then , then is closed.[1][4]

See also[edit]

References[edit]

  1. ^ a b c Rockafellar, R. Tyrrell (1997) [1970]. Convex Analysis. Princeton, NJ: Princeton University Press. pp. 60–76. ISBN 978-0-691-01586-6. 
  2. ^ Borwein, Jonathan; Lewis, Adrian (2006). Convex Analysis and Nonlinear Optimization: Theory and Examples (2 ed.). Springer. ISBN 978-0-387-29570-1. 
  3. ^ a b c d e Zălinescu, Constantin (2002). Convex analysis in general vector spaces. River Edge, NJ: World Scientific Publishing Co., Inc. pp. 6–7. ISBN 981-238-067-1. MR 1921556. 
  4. ^ a b c Kim C. Border. "Sums of sets, etc." (pdf). Retrieved March 7, 2012. 
  5. ^ a b Alfred Auslender; M. Teboulle (2003). Asymptotic cones and functions in optimization and variational inequalities. Springer. pp. 25–80. ISBN 978-0-387-95520-9. 
  6. ^ 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. 
  7. ^ J. Dieudonné (1966). "Sur la séparation des ensembles convexes". Math. Ann. 163.