Radial set

In mathematics, given a linear space $X$, a set $A \subseteq X$ is radial at the point $x_0 \in A$ if for every $x \in X$ there exists a $t_x > 0$ such that for every $t \in [0,t_x]$, $x_0 + tx \in A$.^[1] In set notation, $A$ is radial at the point $x_0 \in A$ if

$\bigcup_{x \in X}\ \bigcap_{t_x > 0}\ \bigcup_{t \in [0,t_x]} \{x_0 + tx\} \subseteq A.$

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.