recession cone

Let C be a convex set in n. If C is bounded, then for any xC, any ray emanating from x will eventually “exit” C (that is, there is a point z on the ray such that zC). If C is unbounded, however, then there exists a point xC, and a ray ρ emanating from x such that ρC. A direction d in C is a point in n such that for any xC, the ray {x+rdr0} is also in C (a subset of C).

The recession cone of C is the set of all directions in C, and is denoted by denoted by 0+C. In other words,


If a convex set C is bounded, then the recession cone of C is pretty useless; it is {0}. The converse is not true, as illustrated by the convex set

C={(x,y)0x<1,y1}{(x,y)0x1, 0y1}.

Clearly, C is not bounded but 0+C={0}. However, if the additional condition that C is closed is imposed, then we recover the converse.

Here are some other examples of recession cones of unbounded convex sets:

  • If C={(x,y)|x|y}, then 0+C=C.

  • If C={(x,y)|x|<y}, then 0+C=C¯, the closure of C.

  • If C={(x,y)|x|ny,n>1}, then 0+C={(0,y)y0}.

Remark. The recession cone of a convex set is convex, and, if the convex set is closed, its recession cone is closed as well.

Title recession cone
Canonical name RecessionCone
Date of creation 2013-03-22 16:20:24
Last modified on 2013-03-22 16:20:24
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 6
Author CWoo (3771)
Entry type Definition
Classification msc 52A20
Classification msc 52A07
Defines direction of a convex set