In mathematics, a subset $\displaystyle{ A \subseteq X }$ of a linear space $\displaystyle{ X }$ is radial at a given point $\displaystyle{ a_0 \in A }$ if for every $\displaystyle{ x \in X }$ there exists a real $\displaystyle{ t_x \gt 0 }$ such that for every $\displaystyle{ t \in [0, t_x], }$ $\displaystyle{ a_0 + t x \in A. }$[1] Geometrically, this means $\displaystyle{ A }$ is radial at $\displaystyle{ a_0 }$ if for every $\displaystyle{ x \in X, }$ there is some (non-degenerate) line segment (depend on $\displaystyle{ x }$) emanating from $\displaystyle{ a_0 }$ in the direction of $\displaystyle{ x }$ that lies entirely in $\displaystyle{ A. }$

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

Relation to absorbing sets

Every absorbing subset is radial at the origin $\displaystyle{ a_0 = 0, }$ 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]

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