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 $t\in [0,t_{x}],$ $a_{0}+tx\in A.$ 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 internal points. 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

• 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

References

• Aliprantis, Charalambos D.; Border, Kim C. (2006). Infinite Dimensional Analysis: A Hitchhiker's Guide (Third ed.). Berlin: Springer Science & Business Media. ISBN 978-3-540-29587-7. OCLC 262692874.
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
• Schechter, Eric (1996). Handbook of Analysis and Its Foundations. San Diego, CA: Academic Press. ISBN 978-0-12-622760-4. OCLC 175294365.