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Mathematical definition

Given a nonempty set A ⊂ X {\displaystyle A\subset X} for some vector space X {\displaystyle X} , then the recession cone recc ⁡ ( A ) {\displaystyle \operatorname {recc} (A)} is given by

recc ⁡ ( A ) = { y ∈ X : ∀ x ∈ A , ∀ λ ≥ 0 : x + λ y ∈ A } . {\displaystyle \operatorname {recc} (A)=\{y\in X:\forall x\in A,\forall \lambda \geq 0:x+\lambda y\in A\}.} ²

If A {\displaystyle A} is additionally a convex set then the recession cone can equivalently be defined by

recc ⁡ ( A ) = { y ∈ X : ∀ x ∈ A : x + y ∈ A } . {\displaystyle \operatorname {recc} (A)=\{y\in X:\forall x\in A:x+y\in A\}.} ³

If A {\displaystyle A} is a nonempty closed convex set then the recession cone can equivalently be defined as

recc ⁡ ( A ) = ⋂ t > 0 t ( A − a ) {\displaystyle \operatorname {recc} (A)=\bigcap _{t>0}t(A-a)} for any choice of a ∈ A . {\displaystyle a\in A.} ⁴

Properties

• If A {\displaystyle A} is a nonempty set then 0 ∈ recc ⁡ ( A ) {\displaystyle 0\in \operatorname {recc} (A)} .
• If A {\displaystyle A} is a nonempty convex set then recc ⁡ ( A ) {\displaystyle \operatorname {recc} (A)} is a convex cone.⁵
• If A {\displaystyle A} is a nonempty closed convex subset of a finite-dimensional Hausdorff space (e.g. R d {\displaystyle \mathbb {R} ^{d}} ), then recc ⁡ ( A ) = { 0 } {\displaystyle \operatorname {recc} (A)=\{0\}} if and only if A {\displaystyle A} is bounded.⁶⁷
• If A {\displaystyle A} is a nonempty set then A + recc ⁡ ( A ) = A {\displaystyle A+\operatorname {recc} (A)=A} where the sum denotes Minkowski addition.

Relation to asymptotic cone

The asymptotic cone for C ⊆ X {\displaystyle C\subseteq X} is defined by

C ∞ = { x ∈ X : ∃ ( t i ) i ∈ I ⊂ ( 0 , ∞ ) , ∃ ( x i ) i ∈ I ⊂ C : t i → 0 , t i x i → x } . {\displaystyle C_{\infty }=\{x\in X:\exists (t_{i})_{i\in I}\subset (0,\infty ),\exists (x_{i})_{i\in I}\subset C:t_{i}\to 0,t_{i}x_{i}\to x\}.} ⁸⁹

By the definition it can easily be shown that recc ⁡ ( C ) ⊆ C ∞ . {\displaystyle \operatorname {recc} (C)\subseteq C_{\infty }.} ¹⁰

In a finite-dimensional space, then it can be shown that C ∞ = recc ⁡ ( C ) {\displaystyle C_{\infty }=\operatorname {recc} (C)} if C {\displaystyle C} is nonempty, closed and convex.¹¹ In infinite-dimensional spaces, then the relation between asymptotic cones and recession cones is more complicated, with properties for their equivalence summarized in.¹²

Sum of closed sets

• Dieudonné's theorem: Let nonempty closed convex sets A , B ⊂ X {\displaystyle A,B\subset X} a locally convex space, if either A {\displaystyle A} or B {\displaystyle B} is locally compact and recc ⁡ ( A ) ∩ recc ⁡ ( B ) {\displaystyle \operatorname {recc} (A)\cap \operatorname {recc} (B)} is a linear subspace, then A − B {\displaystyle A-B} is closed.¹³¹⁴
• Let nonempty closed convex sets A , B ⊂ R d {\displaystyle A,B\subset \mathbb {R} ^{d}} such that for any y ∈ recc ⁡ ( A ) ∖ { 0 } {\displaystyle y\in \operatorname {recc} (A)\backslash \{0\}} then − y ∉ recc ⁡ ( B ) {\displaystyle -y\not \in \operatorname {recc} (B)} , then A + B {\displaystyle A+B} is closed.¹⁵¹⁶

See also

• Barrier cone

References

1. Rockafellar, R. Tyrrell (1997) [1970]. Convex Analysis. Princeton, NJ: Princeton University Press. pp. 60–76. ISBN 978-0-691-01586-6. 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. 978-0-387-29570-1 ↩

3. 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. 981-238-067-1 ↩

4. 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. 981-238-067-1 ↩

5. 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. 981-238-067-1 ↩

6. Rockafellar, R. Tyrrell (1997) [1970]. Convex Analysis. Princeton, NJ: Princeton University Press. pp. 60–76. ISBN 978-0-691-01586-6. 978-0-691-01586-6 ↩

7. 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. 981-238-067-1 ↩

8. Kim C. Border. "Sums of sets, etc" (PDF). Retrieved March 7, 2012. /wiki/Kim_C._Border ↩

9. Alfred Auslender; M. Teboulle (2003). Asymptotic cones and functions in optimization and variational inequalities. Springer. pp. 25–80. ISBN 978-0-387-95520-9. 978-0-387-95520-9 ↩

10. Kim C. Border. "Sums of sets, etc" (PDF). Retrieved March 7, 2012. /wiki/Kim_C._Border ↩

11. Alfred Auslender; M. Teboulle (2003). Asymptotic cones and functions in optimization and variational inequalities. Springer. pp. 25–80. ISBN 978-0-387-95520-9. 978-0-387-95520-9 ↩

12. Zălinescu, Constantin (1993). "Recession cones and asymptotically compact sets". Journal of Optimization Theory and Applications. 77 (1). Springer Netherlands: 209–220. doi:10.1007/bf00940787. ISSN 0022-3239. S2CID 122403313. /wiki/Doi_(identifier) ↩

13. J. Dieudonné (1966). "Sur la séparation des ensembles convexes". Math. Ann.. 163: 1–3. doi:10.1007/BF02052480. S2CID 119742919. /wiki/Doi_(identifier) ↩

14. 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. 981-238-067-1 ↩

15. Rockafellar, R. Tyrrell (1997) [1970]. Convex Analysis. Princeton, NJ: Princeton University Press. pp. 60–76. ISBN 978-0-691-01586-6. 978-0-691-01586-6 ↩

16. Kim C. Border. "Sums of sets, etc" (PDF). Retrieved March 7, 2012. /wiki/Kim_C._Border ↩