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Star domain

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Star domain

In mathematics, a set S in the Euclidean space Rn is called a star domain (or star-convex set, star-shaped set or radially convex set) if there exists an x0 in S such that for all x in S the line segment from x0 to x is in S. This definition is immediately generalizable to any real or complex vector space.

Contents

Intuitively, if one thinks of S as of a region surrounded by a wall, S is a star domain if one can find a vantage point x0 in S from which any point x in S is within line-of-sight.

Examples

  • Any line or plane in Rn is a star domain.
  • A line or a plane with a single point removed is not a star domain.
  • If A is a set in Rn, the set B = { t a : a ∈ A , t ∈ [ 0 , 1 ] } obtained by connecting all points in A 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

  • 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', i.e.: For every dilation ratio r<1, the star domain can be dilated by a ratio r such that the dilated star domain is contained in the original star domain.
  • The union and intersection of two star domains is not necessarily a star domain.
  • A nonempty open star domain S in Rn is diffeomorphic to Rn.

    • References

    Star domain Wikipedia
    (Text) CC BY-SA


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