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 Template:Clarifyalthough not conversely.

Relation to the algebraic interior

The points at which a set is radial are called Template:Em.Template:Sfn 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.Template:Sfn

See also

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References

• Template:Aliprantis Border Infinite Dimensional Analysis A Hitchhiker's Guide Third Edition
• Template:Schaefer Wolff Topological Vector Spaces
• Template:Schechter Handbook of Analysis and Its Foundations