Radial set

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

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 $A \subseteq X$ is radial is equal to the algebraic interior.^[1]^[4]

Relation to absorbing sets

Every absorbing subset is radial at the origin $a_{0} = 0,$ 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]

Notes

1 2 Jaschke & Küchler 2000, p. 23.
↑ Aliprantis & Border 2006, p. 199–200.
↑ John Cook (May 21, 1988). "Separation of Convex Sets in Linear Topological Spaces" (PDF). Retrieved November 14, 2012.
↑ Nikolaĭ Kapitonovich Nikolʹskiĭ (1992). Functional analysis I: linear functional analysis. Springer. ISBN 978-3-540-50584-6.
↑ Schaefer & Wolff 1999, p. 11.

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 ( $μ, ρ$ )-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.

This topology-related article is a stub. You can help Wikipedia by adding missing information.

[FOOTNOTEJaschkeKüchler200023-1] 1 2 Jaschke & Küchler 2000, p. 23.

[FOOTNOTEAliprantisBorder2006199–200-2] Aliprantis & Border 2006, p. 199–200.

[cook-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.

[FOOTNOTESchaeferWolff199911-5] Schaefer & Wolff 1999, p. 11.

[1]

[2]

[3]

[4]

[5]

v t e Topological vector spaces (TVSs)
Basic concepts	Banach space Completeness Continuous linear operator Linear functional Fréchet space Linear map Locally convex space Metrizability Operator topologies Topological vector space Vector space
Main results	Anderson–Kadec Banach–Alaoglu Closed graph theorem F. Riesz's Hahn–Banach (hyperplane separation Vector-valued Hahn–Banach) Open mapping (Banach–Schauder) Bounded inverse Uniform boundedness (Banach–Steinhaus)
Maps	Bilinear operator form Linear map Almost open Bounded Continuous Closed Compact Densely defined Discontinuous Topological homomorphism Functional Linear Bilinear Sesquilinear Norm Seminorm Sublinear function Transpose
Types of sets	Absolutely convex/disk Absorbing/Radial Affine Balanced/Circled Banach disks Bounding points Bounded Complemented subspace Convex Convex cone (subset) Linear cone (subset) Extreme point Pre-compact/Totally bounded Prevalent/Shy Radial Radially convex/Star-shaped Symmetric
Set operations	Affine hull (Relative) Algebraic interior (core) Convex hull Linear span Minkowski addition Polar (Quasi) Relative interior
Types of TVSs	Asplund B-complete/Ptak Banach (Countably) Barrelled BK-space (Ultra-) Bornological Brauner Complete Convenient (DF)-space Distinguished F-space FK-AK space FK-space Fréchet tame Fréchet Grothendieck Hilbert Infrabarreled Interpolation space K-space LB-space LF-space Locally convex space Mackey (Pseudo)Metrizable Montel Nuclear Quasibarrelled Quasi-complete Quasinormed (Polynomially Semi-) Reflexive Riesz Schwartz Semi-complete Smith Stereotype (B Strictly Uniformly) convex (Quasi-) Ultrabarrelled Uniformly smooth Webbed With the approximation property
Category

v t e Convex analysis and variational analysis
Basic concepts	Convex combination Convex function Convex set
Topics (list)	Choquet theory Convex geometry Convex metric space Convex optimization Duality Lagrange multiplier Legendre transformation Locally convex topological vector space Simplex
Maps	Convex conjugate Concave (Closed K- Logarithmically Proper Pseudo- Quasi-) Convex function Invex function Legendre transformation Semi-continuity Subderivative
Main results (list)	Carathéodory's theorem Ekeland's variational principle Fenchel–Moreau theorem Fenchel-Young inequality Jensen's inequality Hermite–Hadamard inequality Krein–Milman theorem Mazur's lemma Shapley–Folkman lemma Robinson–Ursescu Simons Ursescu
Sets	Convex hull (Orthogonally, Pseudo-) Convex set Effective domain Epigraph Hypograph John ellipsoid Lens Radial set/Algebraic interior Zonotope
Series	Convex series related ((cs, lcs)-closed, (cs, bcs)-complete, (lower) ideally convex, (Hx), and (Hwx))
Duality	Dual system Duality gap Strong duality Weak duality
Applications and related	Convexity in economics

Relation to the algebraic interior

Relation to absorbing sets

See also

Notes

References