Radial set explained

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. ^[1] 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 .^[2] The set of all points at which

A \subseteq X is radial is equal to the algebraic interior.^[3]

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.

Notes and References

1. Web site: Coherent Risk Measures, Valuation Bounds, and (\mu, \rho )-Portfolio Optimization. Stefan. Jaschke. Uwe. Küchler. 2000. Humboldt University of Berlin.
2. Web site: Separation of Convex Sets in Linear Topological Spaces . November 14, 2012 . John Cook . May 21, 1988.
3. Book: Nikolaĭ Kapitonovich Nikolʹskiĭ. Functional analysis I: linear functional analysis. 1992. Springer. 978-3-540-50584-6.