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 ^{[clarification needed]}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

1 2 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] 1 2 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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Radial set

Relation to the algebraic interior

Relation to absorbing sets

See also

References