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 mathematics, a set S in the Euclidean space Rn is called a star domain (or star-convex set, star-shaped or radially convex set) if there exists 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.

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[edit]

  • 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= \{ ta : a\in A, t\in[0,1] \}
obtained by connecting any point 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.

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.
  • Any star domain is a contractible set, via a straight-line homotopy. In particular, any star domain is a simply connected set.
  • 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.

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