In mathematics, a subset
of a linear space
is radial at a given point
if for every
there exists a real
such that for every
[1]
Geometrically, this means
is radial at
if for every
there is some (non-degenerate) line segment (depend on
) emanating from
in the direction of
that lies entirely in
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.[3]
The set of all points at which
is radial is equal to the algebraic interior.[1][4]
Relation to absorbing sets
Every absorbing subset is radial at the origin
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
References
- ^ a b Jaschke, Stefan; Küchler, Uwe (2000). "Coherent Risk Measures, Valuation Bounds, and ( μ , ρ {\displaystyle \mu ,\rho }
)-Portfolio Optimization" (PDF). Humboldt University of Berlin.
- ^ John Cook (May 21, 1988). "Separation of Convex Sets in Linear Topological Spaces" (PDF). Retrieved November 14, 2012.
- ^ Nikolaĭ Kapitonovich Nikolʹskiĭ (1992). Functional analysis I: linear functional analysis. Springer. ISBN 978-3-540-50584-6.