Kernel principal component analysis: Difference between revisions
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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.]] | |||
[[Image:Not-star-shaped.svg|right|thumb|An [[annulus (mathematics)|annulus]] is not a star domain.]] | |||
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]]. | |||
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. | |||
==Examples== | |||
* Any line or plane in '''R'''<sup>''n''</sup> 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'''<sup>''n''</sup>, the set | |||
:: <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. | |||
* 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. | |||
==Properties== | |||
* 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. | |||
* Any star domain is a [[contractible_space|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. | |||
* A nonempty open star domain ''S'' in '''R'''<sup>''n''</sup> is [[diffeomorphism|diffeomorphic]] to '''R'''<sup>''n''</sup>. | |||
==See also== | |||
* [[Art gallery problem]] | |||
* [[Star polygon]] — an unrelated term | |||
* [[Star-shaped polygon]] | |||
* [[Balanced set]] | |||
==References== | |||
* 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}} | |||
==External links== | |||
{{commonscat|Star-shaped sets}} | |||
* {{mathworld|urlname=StarConvex|title=Star convex}} | |||
{{Functional Analysis}} | |||
[[Category:Euclidean geometry]] | |||
Revision as of 03:04, 30 January 2014
In mathematics, a set in the Euclidean space Rn is called a star domain (or star-convex set, star-shaped or radially convex set) if there exists x0 in S such that for all x in S the line segment from x0 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 x0 in S from which any point x in S is within line-of-sight.
Examples
- Any line or plane in Rn 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 Rn, the set
- obtained by connecting any point in A to the origin is a star domain.
![{\displaystyle B=\{ta:a\in A,t\in [0,1]\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/70f671ea7702e8863e696d0aa638c5b892188521)
- 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.
Properties
- 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.
- A nonempty open star domain S in Rn is diffeomorphic to Rn.
See also
- Art gallery problem
- Star polygon — an unrelated term
- Star-shaped polygon
- Balanced set
References
- Ian Stewart, David Tall, Complex Analysis. Cambridge University Press, 1983, ISBN 0-521-28763-4, Template:Mr
- C.R. Smith, A characterization of star-shaped sets, American Mathematical Monthly, Vol. 75, No. 4 (April 1968). p. 386, Template:Mr, Template:Jstor
External links
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