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 $t\in [0,t_{x}]$, $x_{0}+tx\in A$. 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. 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.