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 $t \in [0,t_x]$, $x_0 + tx \in A$. 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. The points at which a set is radial are often referred to as internal points.

A set $A \subseteq X$ is absorbing if and only if it is radial at 0. Some authors use the term radial as a synonym for absorbing, i. e. they call a set radial if it is radial at 0.