Radial set

In mathematics, given a linear space , a set is radial at the point if for every there exists a such that for every , .[1] Geometrically, this means is radial at if for every a line segment emanating from in the direction of lies in , where the length of the line segment is required to be non-zero but can depend on .

The set of all points at which is radial is equal to the algebraic interior.[1][2] The points at which a set is radial are often referred to as internal points.[3][4]

A set is absorbing if and only if it is radial at 0.[1] Some authors use the term radial as a synonym for absorbing, i. e. they call a set radial if it is radial at 0.[5]

References

  1. Jaschke, Stefan; Küchler, Uwe (2000). "Coherent Risk Measures, Valuation Bounds, and ( )-Portfolio Optimization". Cite journal requires |journal= (help)
  2. Nikolaĭ Kapitonovich Nikolʹskiĭ (1992). Functional analysis I: linear functional analysis. Springer. ISBN 978-3-540-50584-6.
  3. Aliprantis, C.D.; Border, K.C. (2007). Infinite Dimensional Analysis: A Hitchhiker's Guide (3 ed.). Springer. pp. 199–200. doi:10.1007/3-540-29587-9. ISBN 978-3-540-32696-0.
  4. John Cook (May 21, 1988). "Separation of Convex Sets in Linear Topological Spaces" (pdf). Retrieved November 14, 2012.
  5. Schaefer, Helmuth H. (1971). Topological vector spaces. GTM. 3. New York: Springer-Verlag. ISBN 0-387-98726-6.
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