mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, 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 (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...

, 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 space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...

. It is closely related to the notions of support functions and polar sets.

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 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 ''ℓ'' on ''X'', is known as the support function 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 polar set of ''K''. The set of continuous linear functionals ''ℓ'' for which ''σ''_''K''(''ℓ'') ≤ 0 is known as the (negative) polar cone 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