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 ${\displaystyle t\in [0,t_{x}],}$ ${\displaystyle a_{0}+tx\in A.}$^[1] 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 although not conversely.

Relation to the algebraic interior

The points at which a set is radial are called internal points.^[2][3] The set of all points at which ${\displaystyle A\subseteq X}$ is radial is equal to the algebraic interior.^[1][4]

Relation to absorbing sets

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.^[5]

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.

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