In mathematics, a subset A ⊆ X {\displaystyle A\subseteq X} of a linear space X {\displaystyle X} is radial at a given point a 0 ∈ A {\displaystyle a_{0}\in A} if for every x ∈ X {\displaystyle x\in X} there exists a real t x > 0 {\displaystyle t_{x}>0} such that for every t ∈ [ 0 , t x ] , {\displaystyle t\in [0,t_{x}],} a 0 + t x ∈ A . {\displaystyle a_{0}+tx\in A.} Geometrically, this means A {\displaystyle A} is radial at a 0 {\displaystyle a_{0}} if for every x ∈ X , {\displaystyle x\in X,} there is some (non-degenerate) line segment (depend on x {\displaystyle x} ) emanating from a 0 {\displaystyle a_{0}} in the direction of x {\displaystyle x} that lies entirely in A . {\displaystyle A.}
Relation to the algebraic interior The points at which a set is radial are called internal points. The set of all points at which A ⊆ X {\displaystyle A\subseteq X} is radial is equal to the algebraic interior.
Relation to absorbing sets Every absorbing subset is radial at the origin 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.
See also Absorbing set – Set that can be "inflated" to reach any point Algebraic interior – Generalization of topological interior Minkowski functional – Function made from a set Star domain – Property of point sets in Euclidean spaces
Notes
References Aliprantis, Charalambos D.; Border, Kim C. (2006). Infinite Dimensional Analysis: A Hitchhiker's Guide (Third ed.). Berlin: Springer Science & Business Media. ISBN 978-3-540-29587-7. OCLC 262692874. Jaschke, Stefan; Küchler, Uwe (2000). "Coherent Risk Measures, Valuation Bounds, and ( μ , ρ {\displaystyle \mu ,\rho } )-Portfolio Optimization" (PDF). Humboldt University of Berlin. Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. Schechter, Eric (1996). Handbook of Analysis and Its Foundations. San Diego, CA: Academic Press. ISBN 978-0-12-622760-4. OCLC 175294365.