In geometry, a set S in the Euclidean space is called a star domain (or star-convex set, star-shaped set or radially convex set) if there exists an such that for all the line segment from s0 to s lies in S. This definition is immediately generalizable to any real, or complex, vector space.
Intuitively, if one thinks of S as a region surrounded by a wall, S is a star domain if one can find a vantage point s0 in S from which any point s in S is within line-of-sight. A similar, but distinct, concept is that of a radial set.
- Any line or plane in is a star domain.
- A line or a plane with a single point removed is not a star domain.
- If is a set in the set obtained by connecting all points in 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.
- 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"; that is, for every dilation ratio the star domain can be dilated by a ratio 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 non-empty open star domain in is diffeomorphic to
- Given the set (where ranges over all unit length scalars) is a balanced set whenever is a star shaped at the origin (meaning that and for all and ).
- Absolutely convex set
- Absorbing set – Set that can be "inflated" to reach any point
- Art gallery problem – Mathematical problem
- Balanced set – Construct in functional analysis
- Bounded set (topological vector space) – Generalization of boundedness
- Convex set – In geometry, set that intersects every line into a single line segment
- Star polygon – Regular non-convex polygon
- Symmetric set
- Drummond-Cole, Gabriel C. "What polygons can be shrinked into themselves?". Math Overflow. Retrieved 2 October 2014.
- Ian Stewart, David Tall, Complex Analysis. Cambridge University Press, 1983, ISBN 0-521-28763-4, MR0698076
- C.R. Smith, A characterization of star-shaped sets, American Mathematical Monthly, Vol. 75, No. 4 (April 1968). p. 386, MR0227724, JSTOR 2313423
- Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.