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.