In mathematics, a subset A ⊆ X {\displaystyle A\subseteq X} of a linear space X {\displaystyle X} is radial at a given point a 0 ∈ A {\displaystyle a_{0}\in A} if for every x ∈ X {\displaystyle x\in X} there exists a real t x > 0 {\displaystyle t_{x}>0} such that for every t ∈ [ 0 , t x ] , {\displaystyle t\in [0,t_{x}],} a 0 + t x ∈ A . {\displaystyle a_{0}+tx\in A.} Geometrically, this means A {\displaystyle A} is radial at a 0 {\displaystyle a_{0}} if for every x ∈ X , {\displaystyle x\in X,} there is some (non-degenerate) line segment (depend on x {\displaystyle x}) emanating from a 0 {\displaystyle a_{0}} in the direction of x {\displaystyle x} that lies entirely in A . {\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. The set of all points at which A ⊆ X {\displaystyle A\subseteq X} is radial is equal to the algebraic interior.

Relation to absorbing sets

Every absorbing subset is radial at the origin a 0 = 0 , {\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.

See also