Radial set

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 ${\displaystyle t_{x}>0}$ such that for every ${\displaystyle t\in [0,t_{x}],}$ ${\displaystyle a_{0}+tx\in A.}$^[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.