# Star domain

"Star-shaped" redirects here. For the Blur documentary, see Starshaped.

In mathematics, a set *S* in the Euclidean space **R**^{n} is called a **star domain** (or **star-convex set**, **star-shaped set** or **radially convex set**) if there exists an *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

- Any line or plane in
**R**^{n}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**R**^{n}, 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.^{[1]} - The union and intersection of two star domains is not necessarily a star domain.
- A nonempty open star domain
*S*in**R**^{n}is diffeomorphic to**R**^{n}.

## See also

- Art gallery problem
- Star polygon — an unrelated term
- Balanced set

## References

- ↑ 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, MR 0698076 - C.R. Smith,
*A characterization of star-shaped sets*, American Mathematical Monthly, Vol. 75, No. 4 (April 1968). p. 386, MR 0227724, JSTOR 2313423

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