Radial set

In mathematics, given a linear space X, a set A ⊆ X is radial at the point ${\displaystyle x_{0}\in A}$ if for every x ∈ X there exists a ${\displaystyle t_{x}>0}$ such that for every ${\displaystyle t\in [0,t_{x}]}$, ${\displaystyle x_{0}+tx\in A}$.[1] Geometrically, this means A is radial at ${\displaystyle x_{0}}$ if for every x ∈ X a line segment emanating from ${\displaystyle x_{0}}$ in the direction of x lies in ${\displaystyle A}$, 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 ⊆ X is radial is equal to the algebraic interior.[1][2] The points at which a set is radial are often referred to as internal points.[3][4]

A set A ⊆ X is absorbing if and only if it is radial at 0.[1] Some authors use the term radial as a synonym for absorbing, i. e. they call a set radial if it is radial at 0.[5]