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| [[Image:Star domain.svg|right|thumb|A star domain (equivalently, a star-convex or star-shaped set) is not necessarily [[convex set|convex]] in the ordinary sense.]]
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| [[Image:Not-star-shaped.svg|right|thumb|An [[annulus (mathematics)|annulus]] is not a star domain.]]
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| In [[mathematics]], a [[Set (mathematics)|set]] <math>S</math> in the [[Euclidean space]] '''R'''<sup>''n''</sup> is called a '''star domain''' (or '''star-convex set''', '''star-shaped''' or '''radially convex set''') if there exists ''x''<sub>0</sub> in ''S'' such that for all ''x'' in ''S'' the [[line segment]] from ''x''<sub>0</sub> to ''x'' is in ''S''. This definition is immediately generalizable to any [[real number|real]] or [[complex number|complex]] [[vector space]].
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| 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''<sub>0</sub> in ''S'' from which any point ''x'' in ''S'' is within line-of-sight.
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| ==Examples==
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| * Any line or plane in '''R'''<sup>''n''</sup> is a star domain.
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| * A line or a plane with a single point removed is not a star domain.
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| * If ''A'' is a set in '''R'''<sup>''n''</sup>, the set
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| :: <math>B= \{ ta : a\in A, t\in[0,1] \}</math> | |
| : obtained by connecting any point in ''A'' to the origin is a star domain.
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| * 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.
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| * A [[cross]]-shaped figure is a star domain but is not convex.
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| ==Properties==
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| * The [[closure (topology)|closure]] of a star domain is a star domain, but the [[interior (topology)|interior]] of a star domain is not necessarily a star domain.
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| * Any star domain is a [[contractible_space|contractible]] set, via a straight-line [[homotopy]]. In particular, any star domain is a [[simply connected set]].
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| * The union and intersection of two star domains is not necessarily a star domain.
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| * A nonempty open star domain ''S'' in '''R'''<sup>''n''</sup> is [[diffeomorphism|diffeomorphic]] to '''R'''<sup>''n''</sup>.
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| ==See also== | |
| * [[Art gallery problem]]
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| * [[Star polygon]] — an unrelated term
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| * [[Star-shaped polygon]]
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| * [[Balanced set]]
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| ==References== | |
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| * Ian Stewart, David Tall, ''Complex Analysis''. Cambridge University Press, 1983, ISBN 0-521-28763-4, {{mr|0698076}}
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| * 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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| ==External links==
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| {{commonscat|Star-shaped sets}}
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| * {{mathworld|urlname=StarConvex|title=Star convex}}
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| {{Functional Analysis}}
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| [[Category:Euclidean geometry]]
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