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.}$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.

The points at which a set is radial are called internal points. The set of all points at which ${\displaystyle A\subseteq X}$is radial is equal to the algebraic interior.

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.

• 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

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