"Star-shaped" redirects here. For the Blur documentary, see
Starshaped.
In
geometry, a
set in the
Euclidean space is called a star domain (or star-convex set, star-shaped set or radially convex set) if there exists an such that for all the
line segment from to lies in This definition is immediately generalizable to any
real, or
complex,
vector space.
Intuitively, if one thinks of as a region surrounded by a wall, is a star domain if one can find a vantage point in from which any point in is within line-of-sight. A similar, but distinct, concept is that of a
radial set.
Definition
Given two points and in a vector space (such as
Euclidean space), the
convex hull of is called the closed interval with endpoints and and it is denoted by
where for every vector
A subset of a vector space is said to be star-shaped at if for every the closed interval
A set is star shaped and is called a star domain if there exists some point such that is star-shaped at
A set that is star-shaped at the origin is sometimes called a star set.[1] Such sets are closely related to
Minkowski functionals.
Examples
Any line or plane in is a star domain.
A line or a plane with a single point removed is not a star domain.
If is a set in the set obtained by connecting all points in to the origin is a star domain.
Any
non-emptyconvex set is a star domain. A set is convex if and only if it is a star domain with respect to each point in that set.
A
cross-shaped figure is a star domain but is not convex.
A
star-shaped polygon is a star domain whose boundary is a sequence of connected line segments.
Properties
The
closure of a star domain is a star domain, but the
interior of a star domain is not necessarily a star domain.
Every star domain, and only a star domain, can be "shrunken into itself"; that is, for every dilation ratio the star domain can be dilated by a ratio such that the dilated star domain is contained in the original star domain.[2]
The
union and
intersection of two star domains is not necessarily a star domain.
"Star-shaped" redirects here. For the Blur documentary, see
Starshaped.
In
geometry, a
set in the
Euclidean space is called a star domain (or star-convex set, star-shaped set or radially convex set) if there exists an such that for all the
line segment from to lies in This definition is immediately generalizable to any
real, or
complex,
vector space.
Intuitively, if one thinks of as a region surrounded by a wall, is a star domain if one can find a vantage point in from which any point in is within line-of-sight. A similar, but distinct, concept is that of a
radial set.
Definition
Given two points and in a vector space (such as
Euclidean space), the
convex hull of is called the closed interval with endpoints and and it is denoted by
where for every vector
A subset of a vector space is said to be star-shaped at if for every the closed interval
A set is star shaped and is called a star domain if there exists some point such that is star-shaped at
A set that is star-shaped at the origin is sometimes called a star set.[1] Such sets are closely related to
Minkowski functionals.
Examples
Any line or plane in is a star domain.
A line or a plane with a single point removed is not a star domain.
If is a set in the set obtained by connecting all points in to the origin is a star domain.
Any
non-emptyconvex set is a star domain. A set is convex if and only if it is a star domain with respect to each point in that set.
A
cross-shaped figure is a star domain but is not convex.
A
star-shaped polygon is a star domain whose boundary is a sequence of connected line segments.
Properties
The
closure of a star domain is a star domain, but the
interior of a star domain is not necessarily a star domain.
Every star domain, and only a star domain, can be "shrunken into itself"; that is, for every dilation ratio the star domain can be dilated by a ratio such that the dilated star domain is contained in the original star domain.[2]
The
union and
intersection of two star domains is not necessarily a star domain.