Star domain

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 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.