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 S 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 s0 to s lies in S. This definition is immediately generalizable to any real, or complex, vector space.

Intuitively, if one thinks of S as a region surrounded by a wall, S is a star domain if one can find a vantage point s0 in S from which any point s in S is within line-of-sight. A similar, but distinct, concept is that of a radial set.

Examples[edit]

  • 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.

Properties[edit]

  • 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.[1]
  • 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]

References[edit]

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

External links[edit]