Radial set

In mathematics, a subset $$A \subseteq X$$ of a linear space $$X$$ is radial at a given point $$a_0 \in A$$ if for every $$x \in X$$ there exists a real $$t_x > 0$$ such that for every $$t \in [0, t_x],$$ $$a_0 + t x \in A.$$ Geometrically, this means $$A$$ is radial at $$a_0$$ if for every $$x \in X,$$ there is some (non-degenerate) line segment (depend on $$x$$) emanating from $$a_0$$ in the direction of $$x$$ that lies entirely in $$A.$$

Every radial set is a star domain although not conversely.

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

Relation to absorbing sets
Every absorbing subset is radial at the origin $$a_0 = 0,$$ 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.