In mathematics, especially convex analysis, the recession cone of a set A is a cone containing all vectors such that A recedes in that direction. That is, the set extends outward in all the directions given by the recession cone.
Given a nonempty set A ⊂ X for some vector space X , then the recession cone recc ( A ) is given by
recc ( A ) = { y ∈ X : ∀ x ∈ A , ∀ λ ≥ 0 : x + λ y ∈ A } . If A is additionally a convex set then the recession cone can equivalently be defined by
recc ( A ) = { y ∈ X : ∀ x ∈ A : x + y ∈ A } . If A is a nonempty closed convex set then the recession cone can equivalently be defined as
recc ( A ) = ⋂ t > 0 t ( A − a ) for any choice of
a ∈ A . If A is a nonempty set then 0 ∈ recc ( A ) .If A is a nonempty convex set then recc ( A ) is a convex cone.If A is a nonempty closed convex subset of a finite-dimensional Hausdorff space (e.g. R d ), then recc ( A ) = { 0 } if and only if A is bounded.If A is a nonempty set then A + recc ( A ) = A where the sum denotes Minkowski addition.The asymptotic cone for C ⊆ 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 } . By the definition it can easily be shown that recc ( C ) ⊆ C ∞ .
In a finite-dimensional space, then it can be shown that C ∞ = recc ( C ) if C 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.
Dieudonné's theorem: Let nonempty closed convex sets A , B ⊂ X a locally convex space, if either A or B is locally compact and recc ( A ) ∩ recc ( B ) is a linear subspace, then A − B is closed.Let nonempty closed convex sets A , B ⊂ R d such that for any y ∈ recc ( A ) ∖ { 0 } then − y ∉ recc ( B ) , then A + B is closed.