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 $\mathrm{recc}(A)$ is given by

$\mathrm{recc}(A)=\{y\in X:\mathrm{\forall }x\in A,\mathrm{\forall }\lambda \ge 0:x+\lambda y\in A\}.$

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

$\mathrm{recc}(A)=\{y\in X:\mathrm{\forall }x\in A:x+y\in A\}.$

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

$\mathrm{recc}(A)=\bigcap _{t>0}t(A-a)$ for any choice of $a\in A.$

Properties

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

Relation to asymptotic cone

The asymptotic cone for $C\subseteq X$ is defined by

$C_{\mathrm{\infty }}=\{x\in X:\mathrm{\exists }(t_i{)}_{i\in I}\subset (0,\mathrm{\infty }),\mathrm{\exists }(x_i{)}_{i\in I}\subset C:t_i\to 0,t_i{x}_i\to x\}.$

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

In a finite-dimensional space, then it can be shown that $C_{\mathrm{\infty }}=\mathrm{recc}(C)$ 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 $\mathrm{recc}(A)\cap \mathrm{recc}(B)$ is a linear subspace, then $A-B$ is closed.
• Let nonempty closed convex sets $A,B\subset {ℝ}^d$ such that for any $y\in \mathrm{recc}(A)\mathrm{\setminus }\{0\}$ then $-y\in ̸\mathrm{recc}(B)$, then $A+B$ is closed.

See also

• Barrier cone