mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, 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, t_x

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 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 lies in S. This defini ...

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.

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