Radial set

In mathematics, given a linear space $X$, a set $A\subseteq X$ is radial at the point ${\displaystyle x_{0}\in A}$ if for every $x\in X$ 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 $x_{0}$ if for every $x\in X$ a line segment emanating from $x_{0}$ in the direction of $x$ lies in $A$, 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 $A\subseteq X$ 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 $A\subseteq X$ 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]

References

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