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In mathematics, especially
convex analysis Convex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex minimization, a subdomain of optimization theory. Convex sets A subset C \subseteq X of ...
, the recession cone of a set A is a
cone A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex. A cone is formed by a set of line segments, half-lines, or lines co ...
containing all
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
s 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 In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
X, then the recession cone \operatorname(A) is given by :\operatorname(A) = \. If A is additionally a
convex set In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex ...
then the recession cone can equivalently be defined by :\operatorname(A) = \. If A is a nonempty closed convex set then the recession cone can equivalently be defined as :\operatorname(A) = \bigcap_ t(A - a) for any choice of a \in A.


Properties

* If A is a nonempty set then 0 \in \operatorname(A). * If A is a nonempty convex set then \operatorname(A) is a
convex cone In linear algebra, a ''cone''—sometimes called a linear cone for distinguishing it from other sorts of cones—is a subset of a vector space that is closed under scalar multiplication; that is, is a cone if x\in C implies sx\in C for every . ...
. * If A is a nonempty closed convex subset of a finite-dimensional
Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the many ...
(e.g. \mathbb^d), then \operatorname(A) = \ if and only if A is bounded. * If A is a nonempty set then A + \operatorname(A) = A where the sum denotes
Minkowski addition In geometry, the Minkowski sum (also known as dilation) of two sets of position vectors ''A'' and ''B'' in Euclidean space is formed by adding each vector in ''A'' to each vector in ''B'', i.e., the set : A + B = \. Analogously, the Minkowski ...
.


Relation to asymptotic cone

The asymptotic cone for C \subseteq X is defined by : C_ = \. By the definition it can easily be shown that \operatorname(C) \subseteq C_\infty. In a finite-dimensional space, then it can be shown that C_ = \operatorname(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 In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological ...
, if either A or B is
locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which e ...
and \operatorname(A) \cap \operatorname(B) is a linear subspace, then A - B is closed. * Let nonempty closed convex sets A,B \subset \mathbb^d such that for any y \in \operatorname(A) \backslash \ then -y \not\in \operatorname(B), then A + B is closed.


See also

*
Barrier cone In mathematics, specifically functional analysis, the barrier cone is a cone associated to any non-empty subset of a Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector sp ...


References

{{Reflist Convex analysis