Recession cone

Short description: Set of vectors in convex analysis

In mathematics, especially convex analysis, the recession cone of a set [math]\displaystyle{ A }[/math] is a cone containing all vectors such that [math]\displaystyle{ A }[/math] 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 [math]\displaystyle{ A \subset X }[/math] for some vector space [math]\displaystyle{ X }[/math], then the recession cone [math]\displaystyle{ \operatorname{recc}(A) }[/math] is given by

[math]\displaystyle{ \operatorname{recc}(A) = \{y \in X: \forall x \in A, \forall \lambda \geq 0: x + \lambda y \in A\}. }[/math]

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

[math]\displaystyle{ \operatorname{recc}(A) = \{y \in X: \forall x \in A: x + y \in A\}. }[/math]

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

[math]\displaystyle{ \operatorname{recc}(A) = \bigcap_{t \gt 0} t(A - a) }[/math] for any choice of [math]\displaystyle{ a \in A. }[/math]

Properties

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

Relation to asymptotic cone

The asymptotic cone for [math]\displaystyle{ C \subseteq X }[/math] is defined by

[math]\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\}. }[/math]

By the definition it can easily be shown that [math]\displaystyle{ \operatorname{recc}(C) \subseteq C_\infty. }[/math]

In a finite-dimensional space, then it can be shown that [math]\displaystyle{ C_{\infty} = \operatorname{recc}(C) }[/math] if [math]\displaystyle{ C }[/math] 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 [math]\displaystyle{ A,B \subset X }[/math] a locally convex space, if either [math]\displaystyle{ A }[/math] or [math]\displaystyle{ B }[/math] is locally compact and [math]\displaystyle{ \operatorname{recc}(A) \cap \operatorname{recc}(B) }[/math] is a linear subspace, then [math]\displaystyle{ A - B }[/math] is closed.
• Let nonempty closed convex sets [math]\displaystyle{ A,B \subset \mathbb{R}^d }[/math] such that for any [math]\displaystyle{ y \in \operatorname{recc}(A) \backslash \{0\} }[/math] then [math]\displaystyle{ -y \not\in \operatorname{recc}(B) }[/math], then [math]\displaystyle{ A + B }[/math] is closed.

See also

• Barrier cone