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 ^{[clarification needed]}although not conversely.

Relation to the algebraic interior

The points at which a set is radial are called internal points.^[2][3] The set of all points at which ${\displaystyle A\subseteq X}$ is radial is equal to the algebraic interior.^[1][4]

Relation to absorbing sets

Every absorbing subset is radial at the origin ${\displaystyle a_{0}=0,}$ and if the vector space is real then the converse also holds. That is, a subset of a real vector space is absorbing if and only if it is radial at the origin. Some authors use the term radial as a synonym for absorbing.^[5]

See also

• Absorbing set – Set that can be "inflated" to reach any point
• Algebraic interior – Generalization of topological interior
• Minkowski functional – Function made from a set
• Star domain – Property of point sets in Euclidean spaces