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 , 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''.

See also

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References

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Convex analysis
Functional analysis
Linear algebra
Topology