Star domain

Star domain

In mathematics, a set S in the Euclidean space R"n" is called a star domain (or star-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 fence, 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

* Any line or plane in R"n" is a star domain.
* A line or a plane without a point is not a star domain.
* If "A" is a set in R"n", the set :: B= { ta : ain A, tin [0,1] } : obtained by connecting any point in "A" to the origin is a star domain.

Properties

* Any non-empty convex set is a star domain. A set is convex if and only if it is a star domain in 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.

ee also

* Art gallery problem
* Star polygon — an unrelated term
* Star-shaped polygon

References

* Ian Stewart, David Tall, "Complex Analysis". Cambridge University Press, 1983. ISBN 0-521-28763-4.

* C.R. Smith, "A characterization of Star-shaped sets", American Mathematical Monthly, Vol. 75, No. 4 (April 1968). pp. 386.

External links

*


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