Recession cone

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.

Mathematical definition
Given a nonempty set $$A \subset X$$ for some vector space $$X$$, then the recession cone $$\operatorname{recc}(A)$$ is given by
 * $$\operatorname{recc}(A) = \{y \in X: \forall x \in A, \forall \lambda \geq 0: x + \lambda y \in A\}.$$

If $$A$$ is additionally a convex set then the recession cone can equivalently be defined by
 * $$\operatorname{recc}(A) = \{y \in X: \forall x \in A: x + y \in A\}.$$

If $$A$$ is a nonempty closed convex set then the recession cone can equivalently be defined as
 * $$\operatorname{recc}(A) = \bigcap_{t > 0} t(A - a)$$ for any choice of $$a \in A.$$

Properties

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

Relation to asymptotic cone
The asymptotic cone for $$C \subseteq X$$ is defined by
 * $$C_{\infty} = \{x \in X: \exists (t_i)_{i \in I} \subset (0,\infty), \exists (x_i)_{i \in I} \subset C: t_i \to 0, t_i x_i \to x\}.$$

By the definition it can easily be shown that $$\operatorname{recc}(C) \subseteq C_\infty.$$

In a finite-dimensional space, then it can be shown that $$C_{\infty} = \operatorname{recc}(C)$$ 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.

Sum of closed sets

 * Dieudonné's theorem: Let nonempty closed convex sets $$A,B \subset X$$ a locally convex space, if either $$A$$ or $$B$$ is locally compact and $$\operatorname{recc}(A) \cap \operatorname{recc}(B)$$ is a linear subspace, then $$A - B$$ is closed.
 * Let nonempty closed convex sets $$A,B \subset \mathbb{R}^d$$ such that for any $$y \in \operatorname{recc}(A) \backslash \{0\}$$ then $$-y \not\in \operatorname{recc}(B)$$, then $$A + B$$ is closed.