Radial set

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In mathematics, given a linear space X, a set A \subseteq X is radial at the point x_0 \in A if for every x \in X there exists a t_x > 0 such that for every t \in [0,t_x], x_0 + tx \in A.[1] In set notation, A is radial at the point x_0 \in A if

\bigcup_{x \in X}\ \bigcap_{t_x > 0}\ \bigcup_{t \in [0,t_x]} \{x_0 + tx\} \subseteq A.

The set of all points at which A \subseteq X 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 A \subseteq X 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[edit]

  1. ^ a b c Jaschke, Stefan; Küchler, Uwe (2000). Coherent Risk Measures, Valuation Bounds, and (\mu,\rho)-Portfolio Optimization. 
  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.