Star domain

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A star domain (equivalently, a star-convex or star-shaped set) is not necessarily convex in the ordinary sense.
An annulus is not a star domain.

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.


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 closed related to Minkowski functionals.


  • 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-empty convex set is a star domain. A set is convex if and only if it is a star domain with respect to any 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.


  • 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 is a contractible set, via a straight-line homotopy. In particular, any star domain is a simply connected set.
  • 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.
  • A non-empty open star domain in is diffeomorphic to
  • Given the set (where ranges over all unit length scalars) is a balanced set whenever is a star shaped at the origin (meaning that and for all and ).

See also[edit]


  1. ^ Schechter 1996, p. 303.
  2. ^ Drummond-Cole, Gabriel C. "What polygons can be shrinked into themselves?". Math Overflow. Retrieved 2 October 2014.

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