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Radial set

{{Short description|Topological set}}

In mathematics, a subset A \subseteq X of a linear space X is radial at a given point a_0 \in A if for every x \in X there exists a real t_x > 0 such that for every t \in [0, t_x], a_0 + t x \in A.{{cite web|title=Coherent Risk Measures, Valuation Bounds, and (\mu, \rho)-Portfolio Optimization|first1=Stefan|last1=Jaschke|first2=Uwe|last2=Küchler|year=2000|url=https://edoc.hu-berlin.de/bitstream/handle/18452/4328/64.pdf|publisher=Humboldt University of Berlin}}

Geometrically, this means A is radial at a_0 if for every x \in X, there is some (non-degenerate) line segment (depend on x) emanating from a_0 in the direction of x that lies entirely in A.

Every radial set is a star domain {{clarify|date=February 2025}}although not conversely.

Relation to the algebraic interior

The points at which a set is radial are called {{em|internal points}}.{{sfn|Aliprantis|Border|2006|p=199–200}}{{cite web|url=http://www.johndcook.com/SeparationOfConvexSets.pdf | accessdate=November 14, 2012 |title=Separation of Convex Sets in Linear Topological Spaces |author=John Cook |date=May 21, 1988}}

The set of all points at which A \subseteq X is radial is equal to the algebraic interior.{{cite book|author=Nikolaĭ Kapitonovich Nikolʹskiĭ|title=Functional analysis I: linear functional analysis|year=1992|publisher=Springer|isbn=978-3-540-50584-6}}

Relation to absorbing sets

Every absorbing subset is radial at the origin a_0 = 0, 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.{{sfn|Schaefer|Wolff|1999|p=11}}

See also

  • {{annotated link|Absorbing set}}
  • {{annotated link|Algebraic interior}}
  • {{annotated link|Minkowski functional}}
  • {{annotated link|Star domain}}

References

{{reflist}}

  • {{Aliprantis Border Infinite Dimensional Analysis A Hitchhiker's Guide Third Edition}}
  • {{Schaefer Wolff Topological Vector Spaces|edition=2}}
  • {{Schechter Handbook of Analysis and Its Foundations}}

{{Functional analysis}}

{{Topological vector spaces}}

{{Convex analysis and variational analysis}}

{{topology-stub}}

Category:Convex analysis

Category:Functional analysis

Category:Linear algebra

Category:Topology