Radial set

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In mathematics, a subset AX of a linear space X is radial at a given point a0A if for every xX there exists a real tx>0 such that for every t[0,tx], a0+txA.[1] Geometrically, this means A is radial at a0 if for every xX, there is some (non-degenerate) line segment (depend on x) emanating from a0 in the direction of x that lies entirely in 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 AX is radial is equal to the algebraic interior.[1][4]

Relation to absorbing sets

Every absorbing subset is radial at the origin a0=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. 1.0 1.1 Jaschke, Stefan; Küchler, Uwe (2000). Coherent Risk Measures, Valuation Bounds, and (μ,ρ)-Portfolio Optimization. 
  2. Aliprantis & Border 2006, p. 199–200.
  3. John Cook (May 21, 1988). "Separation of Convex Sets in Linear Topological Spaces". http://www.johndcook.com/SeparationOfConvexSets.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.