Portal:Mathematics/Featured picture/2011 10

2011-10-27T14:53:01Z

68.232.253.249: Undid revision 457591990 by 121.1.38.226 (talk)

In [[mathematics]], given a [[linear space]] <math>X</math>, a set <math>A \subseteq X</math> is '''radial''' at the point <math>x_0 \in A</math> if for every <math>x \in X</math> there exists a <math>t_x > 0</math> such that for every <math>t \in [0,t_x]</math>, <math>x_0 + tx \in A</math>.<ref name="coherent">{{cite journal|title=Coherent Risk Measures, Valuation Bounds, and (<math>\mu,\rho</math>)-Portfolio Optimization|first1=Stefan|last1=Jaschke|first2=Uwe|last2=Küchler|year=2000}}</ref> In set notation, <math>A</math> is radial at the point <math>x_0 \in A</math> if
:<math>\bigcup_{x \in X}\ \bigcap_{t_x > 0}\ \bigcup_{t \in [0,t_x]} \{x_0 + tx\} \subseteq A.</math>

The set of all points at which <math>A \subseteq X</math> is radial is equal to the [[algebraic interior]].<ref name="coherent" /><ref>{{cite book|author=Nikolaĭ Kapitonovich Nikolʹskiĭ|title=Functional analysis I: linear functional analysis|year=1992|publisher=Springer|isbn=978-3-540-50584-6}}</ref> The points at which a set is radial are often referred to as internal points.<ref name="aliprantis+border">{{cite book|last=Aliprantis|first=C.D.|last2=Border|first2=K.C.|title=Infinite Dimensional Analysis: A Hitchhiker's Guide|edition=3|publisher=Springer|year=2007|isbn=978-3-540-32696-0|doi=10.1007/3-540-29587-9|pages=199-200}}</ref><ref name="cook">{{cite web|url=http://www.johndcook.com/SeparationOfConvexSets.pdf | accessdate=November 14, 2012 |format=pdf |title=Separation of Convex Sets in Linear Topological Spaces |author=John Cook |date=May 21, 1988}}</ref>

A set <math>A \subseteq X</math> is [[absorbing set|absorbing]] if and only if it is radial at 0.<ref name="coherent" /> Some authors use the term ''radial'' as a synonym for ''absorbing'', i. e. they call a set radial if it is radial at 0.<ref name="schaefer">{{cite book | last = Schaefer | first = Helmuth H.  | year = 1971 | title = Topological vector spaces | series=[[Graduate Texts in Mathematics|GTM]] | volume=3 | publisher = Springer-Verlag | location = New York | isbn = 0-387-98726-6}}</ref>

== References ==
{{Reflist}}

{{Functional Analysis}}

[[Category:Topology]]

{{topology-stub}}

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Portal:Mathematics/Featured picture/2011 10