mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, specifically

functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics ...

, the barrier cone is a

cone In geometry, a cone is a three-dimensional figure that tapers smoothly from a flat base (typically a circle) to a point not contained in the base, called the '' apex'' or '' vertex''. A cone is formed by a set of line segments, half-lines ...

associated to any non-empty subset of a

Banach space In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and ...

. It is closely related to the notions of

support function In mathematics, the support function ''h'A'' of a non-empty closed convex set ''A'' in \mathbb^n describes the (signed) distances of supporting hyperplanes of ''A'' from the origin. The support function is a convex function on \mathbb^n. Any ...

s and

polar set In functional and convex analysis, and related disciplines of mathematics, the polar set A^ is a special convex set associated to any subset A of a vector space X, lying in the dual space X^. The bipolar of a subset is the polar of A^\circ, but ...

Definition

Let ''X'' be a Banach space and let ''K'' be a non-empty subset of ''X''. The barrier cone of ''K'' is the subset ''b''(''K'') of ''X''^∗, the

continuous dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V,'' together with the vector space structure of pointwise addition and scalar multiplication by const ...

of ''X'', defined by :

b(K) := \left\.

Related notions

The function :

\sigma_ \colon \ell \mapsto \sup_ \langle \ell, x \rangle,

defined for each

continuous linear functional In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces. An operator between two normed spaces is a bounded linear ...

''ℓ'' on ''X'', is known as the

of the set ''K''; thus, the barrier cone of ''K'' is precisely the set of continuous linear functionals ''ℓ'' for which ''σ''_''K''(''ℓ'') is finite. The set of continuous linear functionals ''ℓ'' for which ''σ''_''K''(''ℓ'') ≤ 1 is known as the

of ''K''. The set of continuous linear functionals ''ℓ'' for which ''σ''_''K''(''ℓ'') ≤ 0 is known as the (negative)

polar cone Dual cone and polar cone are closely related concepts in convex analysis, a branch of mathematics. Dual cone In a vector space The dual cone ''C'' of a subset ''C'' in a linear space ''X'' over the reals, e.g. Euclidean space R''n'', wit ...

of ''K''. Clearly, both the polar set and the negative polar cone are subsets of the barrier cone.

References

* {{cite book , last = Aubin , first = Jean-Pierre , author2=Frankowska, Hélène, author2-link=Hélène Frankowska , title = Set-Valued Analysis , year = 2009 , publisher = Birkhäuser Boston Inc. , location = Boston, MA , isbn = 978-0-8176-4847-3 , pages = xx+461 , edition = Reprint of the 1990 , mr = 2458436 Functional analysis