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 $t\in [0,t_{x}],$ $a_{0}+tx\in A.$ ^[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 $A.$

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 $A\subseteq X$ 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]

References

^ ^a ^b Jaschke, Stefan; Küchler, Uwe (2000). "Coherent Risk Measures, Valuation Bounds, and ( $\mu ,\rho$ )-Portfolio Optimization" (PDF). Humboldt University of Berlin.
^ 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.

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.
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.

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[coherent-1] Jaschke, Stefan; Küchler, Uwe (2000). "Coherent Risk Measures, Valuation Bounds, and ( $\mu ,\rho$ )-Portfolio Optimization" (PDF). Humboldt University of Berlin.

[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.

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v t e Topological vector spaces (TVSs)
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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
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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
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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

References