Recession cone

In mathematics, especially convex analysis, the recession cone of a set ${\displaystyle A}${\displaystyle A} is a cone containing all vectors such that ${\displaystyle A}${\displaystyle A} recedes in that direction. That is, the set extends outward in all the directions given by the recession cone.^[1]

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

${\displaystyle \operatorname {recc} (A)=\{y\in X:\forall x\in A,\forall \lambda \geq 0:x+\lambda y\in A\}.}${\displaystyle \operatorname {recc} (A)=\{y\in X:\forall x\in A,\forall \lambda \geq 0:x+\lambda y\in A\}.}^[2]

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

${\displaystyle \operatorname {recc} (A)=\{y\in X:\forall x\in A:x+y\in A\}.}${\displaystyle \operatorname {recc} (A)=\{y\in X:\forall x\in A:x+y\in A\}.}^[3]

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

${\displaystyle \operatorname {recc} (A)=\bigcap _{t>0}t(A-a)}${\displaystyle \operatorname {recc} (A)=\bigcap _{t>0}t(A-a)} for any choice of ${\displaystyle a\in A.}${\displaystyle a\in A.}^[3]

• If ${\displaystyle A}${\displaystyle A} is a nonempty set then ${\displaystyle 0\in \operatorname {recc} (A)}${\displaystyle 0\in \operatorname {recc} (A)}.
• If ${\displaystyle A}${\displaystyle A} is a nonempty convex set then ${\displaystyle \operatorname {recc} (A)}${\displaystyle \operatorname {recc} (A)} is a convex cone.^[3]
• If ${\displaystyle A}${\displaystyle A} is a nonempty closed convex subset of a finite-dimensional Hausdorff space (e.g. ${\displaystyle \mathbb {R} ^{d}}${\displaystyle \mathbb {R} ^{d}}), then ${\displaystyle \operatorname {recc} (A)=\{0\}}${\displaystyle \operatorname {recc} (A)=\{0\}} if and only if ${\displaystyle A}${\displaystyle A} is bounded.^{[1] [3]}
• If ${\displaystyle A}${\displaystyle A} is a nonempty set then ${\displaystyle A+\operatorname {recc} (A)=A}${\displaystyle A+\operatorname {recc} (A)=A} where the sum denotes Minkowski addition.

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

${\displaystyle C_{\infty }=\{x\in X:\exists (t_{i})_{i\in I}\subset (0,\infty ),\exists (x_{i})_{i\in I}\subset C:t_{i}\to 0,t_{i}x_{i}\to x\}.}${\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\}.}^{[4] [5]}

By the definition it can easily be shown that ${\displaystyle \operatorname {recc} (C)\subseteq C_{\infty }.}${\displaystyle \operatorname {recc} (C)\subseteq C_{\infty }.}^[4]

In a finite-dimensional space, then it can be shown that ${\displaystyle C_{\infty }=\operatorname {recc} (C)}${\displaystyle C_{\infty }=\operatorname {recc} (C)} if ${\displaystyle C}${\displaystyle C} is nonempty, closed and convex.^[5] In infinite-dimensional spaces, then the relation between asymptotic cones and recession cones is more complicated, with properties for their equivalence summarized in.^[6]

• Dieudonné's theorem: Let nonempty closed convex sets ${\displaystyle A,B\subset X}${\displaystyle A,B\subset X} a locally convex space, if either ${\displaystyle A}${\displaystyle A} or ${\displaystyle B}${\displaystyle B} is locally compact and ${\displaystyle \operatorname {recc} (A)\cap \operatorname {recc} (B)}${\displaystyle \operatorname {recc} (A)\cap \operatorname {recc} (B)} is a linear subspace, then ${\displaystyle A-B}${\displaystyle A-B} is closed.^{[7] [3]}
• Let nonempty closed convex sets ${\displaystyle A,B\subset \mathbb {R} ^{d}}${\displaystyle A,B\subset \mathbb {R} ^{d}} such that for any ${\displaystyle y\in \operatorname {recc} (A)\backslash \{0\}}${\displaystyle y\in \operatorname {recc} (A)\backslash \{0\}} then ${\displaystyle -y\not \in \operatorname {recc} (B)}${\displaystyle -y\not \in \operatorname {recc} (B)}, then ${\displaystyle A+B}${\displaystyle A+B} is closed.^{[1] [4]}

• Barrier cone

• Convex analysis

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