Radial set

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

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 A ⊆ X {\displaystyle A\subseteq X} {\displaystyle A\subseteq X} is radial is equal to the algebraic interior.[1][4]

Relation to absorbing sets

Every absorbing subset is radial at the origin a 0 = 0 , {\displaystyle a_{0}=0,} {\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]

See also

References

  1. Jaschke, Stefan; Küchler, Uwe (2000). "Coherent Risk Measures, Valuation Bounds, and ( μ , ρ {\displaystyle \mu ,\rho } {\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.