"Star-shaped" redirects here. For the Blur documentary, see
A star domain (equivalently, a star-convex or star-shaped set) is not necessarily
in the ordinary sense.
mathematics, a set in the Euclidean space R is called a n star domain (or star-convex set, star-shaped or radially convex set) if there exists x 0 in S such that for all x in S the line segment from x 0 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 x 0 in S from which any point x in S is within line-of-sight.
Examples [ edit ]
Any line or plane in
R is a star domain. n A line or a plane with a single point removed is not a star domain.
A is a set in R , the set n
obtained by connecting any point in
A to the origin is a star domain.
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 ]
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 'shrinked 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. [1 ] The union and intersection of two star domains is not necessarily a star domain.
A nonempty open star domain
S in R is n diffeomorphic to R . n
See also [ edit ]
References [ edit ]
External links [ edit ]