In mathematics, a subset

A \subseteq X

of a

linear space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...

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

, t_x The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...

a_0 + t x \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 In geometry, a Set (mathematics), set S in the Euclidean space \R^n is called a star domain (or star-convex set, star-shaped set or radially convex set) if there exists an s_0 \in S such that for all s \in S, the line segment from s_0 to s lie ...

although not conversely.

Relation to the algebraic interior

The points at which a set is radial are called . The set of all points at which

A \subseteq X

is radial is equal to the

algebraic interior In functional analysis, a branch of mathematics, the algebraic interior or radial kernel of a subset of a vector space is a refinement of the concept of the interior. Definition Assume that A is a subset of a vector space X. The ''algebraic i ...

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''.

References

* * * {{topology-stub Convex analysis Functional analysis Linear algebra Topology

Relation to the algebraic interior

Relation to absorbing sets

See also

References