Recession cone

In mathematics, especially convex analysis, the recession cone of a set A {displaystyle A} is a cone containing all vectors such that A {displaystyle A} recedes in that direction. That is, the set extends outward in all the directions given by the recession cone. In mathematics, especially convex analysis, the recession cone of a set A {displaystyle A} is a cone containing all vectors such that A {displaystyle 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 {displaystyle Asubset X} for some vector space X {displaystyle X} , then the recession cone recc ⁡ ( A ) {displaystyle operatorname {recc} (A)} is given by If A {displaystyle A} is additionally a convex set then the recession cone can equivalently be defined by If A {displaystyle A} is a nonempty closed convex set then the recession cone can equivalently be defined as The asymptotic cone for C ⊆ X {displaystyle Csubseteq X} is defined by By the definition it can easily be shown that recc ⁡ ( C ) ⊆ C ∞ . {displaystyle operatorname {recc} (C)subseteq C_{infty }.} In a finite-dimensional space, then it can be shown that C ∞ = recc ⁡ ( C ) {displaystyle C_{infty }=operatorname {recc} (C)} if C {displaystyle 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.