Radial set
In mathematics, a subset [math]\displaystyle{ A \subseteq X }[/math] of a linear space [math]\displaystyle{ X }[/math] is radial at a given point [math]\displaystyle{ a_0 \in A }[/math] if for every [math]\displaystyle{ x \in X }[/math] there exists a real [math]\displaystyle{ t_x \gt 0 }[/math] such that for every [math]\displaystyle{ t \in [0, t_x], }[/math] [math]\displaystyle{ a_0 + t x \in A. }[/math][1] Geometrically, this means [math]\displaystyle{ A }[/math] is radial at [math]\displaystyle{ a_0 }[/math] if for every [math]\displaystyle{ x \in X, }[/math] there is some (non-degenerate) line segment (depend on [math]\displaystyle{ x }[/math]) emanating from [math]\displaystyle{ a_0 }[/math] in the direction of [math]\displaystyle{ x }[/math] that lies entirely in [math]\displaystyle{ A. }[/math]
Every radial set is a star domain although not conversely.
Relation to the algebraic interior
The points at which a set is radial are called internal points.[2][3] The set of all points at which [math]\displaystyle{ A \subseteq X }[/math] is radial is equal to the algebraic interior.[1][4]
Relation to absorbing sets
Every absorbing subset is radial at the origin [math]\displaystyle{ a_0 = 0, }[/math] and if the vector space is real then the converse also holds. That is, a subset of a real vector space is absorbing if and only if it is radial at the origin. Some authors use the term radial as a synonym for absorbing.[5]
See also
- Absorbing set – Set that can be "inflated" to reach any point
- Algebraic interior – Generalization of topological interior
- Minkowski functional – Function made from a set
- Star domain – Property of point sets in Euclidean spaces
References
- ↑ 1.0 1.1 Jaschke, Stefan; Küchler, Uwe (2000). Coherent Risk Measures, Valuation Bounds, and ([math]\displaystyle{ \mu, \rho }[/math])-Portfolio Optimization.
- ↑ Aliprantis & Border 2006, p. 199–200.
- ↑ John Cook (May 21, 1988). "Separation of Convex Sets in Linear Topological Spaces". http://www.johndcook.com/SeparationOfConvexSets.pdf. Retrieved November 14, 2012.
- ↑ Nikolaĭ Kapitonovich Nikolʹskiĭ (1992). Functional analysis I: linear functional analysis. Springer. ISBN 978-3-540-50584-6.
- ↑ Schaefer & Wolff 1999, p. 11.
- Template:Aliprantis Border Infinite Dimensional Analysis A Hitchhiker's Guide Third Edition
- Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
- Schechter, Eric (1996). Handbook of Analysis and Its Foundations. San Diego, CA: Academic Press. ISBN 978-0-12-622760-4. OCLC 175294365.
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