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Radial set - Wikipedia

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In mathematics, a subset {\displaystyle A\subseteq X} of a linear space {\displaystyle X} is radial at a given point {\displaystyle a_{0}\in A} if for every {\displaystyle x\in X} there exists a real {\displaystyle t_{x}>0} such that for every {\displaystyle t\in [0,t_{x}],} {\displaystyle a_{0}+tx\in A.}[1] Geometrically, this means {\displaystyle A} is radial at {\displaystyle a_{0}} if for every {\displaystyle x\in X,} there is some (non-degenerate) line segment (depend on {\displaystyle x}) emanating from {\displaystyle a_{0}} in the direction of {\displaystyle x} that lies entirely in {\displaystyle A.}

Every radial set is a star domain [clarification needed]although not conversely.

Relation to the algebraic interior

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The points at which a set is radial are called internal points.[2][3] The set of all points at which {\displaystyle A\subseteq X} is radial is equal to the algebraic interior.[1][4]

Relation to absorbing sets

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Every absorbing subset is radial at the origin {\displaystyle 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.[5]

  1. ^ a b Jaschke, Stefan; Küchler, Uwe (2000). "Coherent Risk Measures, Valuation Bounds, and ({\displaystyle \mu ,\rho })-Portfolio Optimization" (PDF). Humboldt University of Berlin.
  2. ^ Aliprantis & Border 2006, p. 199–200.
  3. ^ John Cook (May 21, 1988). "Separation of Convex Sets in Linear Topological Spaces" (PDF). Retrieved November 14, 2012.
  4. ^ Nikolaĭ Kapitonovich Nikolʹskiĭ (1992). Functional analysis I: linear functional analysis. Springer. ISBN 978-3-540-50584-6.
  5. ^ Schaefer & Wolff 1999, p. 11.