Radial set

In mathematics, given a linear space X, a set A ⊆ X is radial at the point ${\displaystyle x_{0}\in A}$ if for every x ∈ X there exists a ${\displaystyle t_{x}>0}$ such that for every ${\displaystyle t\in [0,t_{x}]}$, ${\displaystyle x_{0}+tx\in A}$.[1] Geometrically, this means A is radial at ${\displaystyle x_{0}}$ if for every x ∈ X a line segment emanating from ${\displaystyle x_{0}}$ in the direction of x lies in ${\displaystyle A}$, where the length of the line segment is required to be non-zero but can depend on x.

The set of all points at which A ⊆ 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 ⊆ 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]

See also

• Absorbing set

References

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