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.

Contents

Examples

Properties

See also

References

External links

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 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 simply connected set.

★ The union and intersection of two star domains is not necessarily a star domain.

★ Any star domain S in 'R'^''n'' is diffeomorphic to 'R'^''n''.

See 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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