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