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{{Short description|Property of point sets in Euclidean spaces}}
{{redirect|Star-shaped|the Blur documentary|Starshaped}}
[[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 [[geometry]], a [[Set (mathematics)|set]] <math>S</math> in the [[Euclidean space]] <math>\R^n</math> is called a '''star domain''' (or '''star-convex set''', '''star-shaped set'''<ref name=":0">{{Cite journal |last=Braga de Freitas |first=Sinval |last2=Orrillo |first2=Jaime |last3=Sosa |first3=Wilfredo |date=2020-11-01 |title=From Arrow–Debreu condition to star shape preferences |url=https://www.tandfonline.com/doi/full/10.1080/02331934.2019.1576664 |journal=Optimization |language=en |volume=69 |issue=11 |pages=2405–2419 |doi=10.1080/02331934.2019.1576664 |issn=0233-1934}}</ref>  or '''radially convex set''') if there exists an <math>s_0 \in S</math> such that for all <math>s \in S,</math> the [[line segment]] from <math>s_0</math> to <math>s</math> lies in <math>S.</math> This definition is immediately generalizable to any [[Real number|real]], or [[Complex number|complex]], [[vector space]].

Intuitively, if one thinks of <math>S</math> as a region surrounded by a wall, <math>S</math> is a star domain if one can find a vantage point <math>s_0</math> in <math>S</math> from which any point <math>s</math> in <math>S</math> is within line-of-sight. A similar, but distinct, concept is that of a [[radial set]].

==Definition==

Given two points <math>x</math> and <math>y</math> in a vector space <math>X</math> (such as [[Euclidean space]] <math>\R^n</math>), the [[convex hull]] of <math>\{x, y\}</math> is called the {{em|closed interval with endpoints <math>x</math> and <math>y</math>}} and it is denoted by
<math display=block>\left[x, y\right] ~:=~ \left\{t x + (1 - t) y : 0 \leq t \leq 1\right\} ~=~ x + (y - x) [0, 1],</math>
where <math>z [0, 1] := \{z t : 0 \leq t \leq 1\}</math> for every vector <math>z.</math>

A subset <math>S</math> of a vector space <math>X</math> is said to be {{em|star-shaped at}} <math>s_0 \in S</math> if for every <math>s \in S,</math> the closed interval
<math>\left[s_0, s\right] \subseteq S.</math>
A set <math>S</math>  is {{em|star shaped}} and is called a {{em|star domain}} if there exists some point <math>s_0 \in S</math> such that <math>S</math> is star-shaped at <math>s_0.</math>

A set that is star-shaped at the origin is sometimes called a {{em|star set}}.{{sfn|Schechter|1996|p=303}} Such sets are closely related to [[Minkowski functional]]s.

==Examples==

* Any line or plane in <math>\R^n</math> is a star domain.
* A line or a plane with a single point removed is not a star domain.
* If <math>A</math> is a set in <math>\R^n,</math> the set <math>B = \{t a : a \in A, t \in [0, 1]\}</math> obtained by connecting all points in <math>A</math> to the origin is a star domain.
* 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==

* '''Convexity''': any [[Empty set|non-empty]] [[convex set]] is a star domain. A set is convex if and only if it is a star domain with respect to each point in that set.
* '''Closure and interior:''' 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.
* '''Contraction''': Every star domain is a [[Contractible space|contractible]] set, via a [[Homotopy#Examples|straight-line homotopy]].  In particular, any star domain is a [[simply connected]] set.
* '''Shrinking''': Every star domain, and only a star domain, can be "shrunken into itself"; that is, for every dilation ratio <math>r < 1,</math> the star domain can be dilated by a ratio <math>r</math> such that the dilated star domain is contained in the original star domain.<ref>{{cite web|last1=Drummond-Cole|first1=Gabriel C.|title=What polygons can be shrinked into themselves?|url=https://mathoverflow.net/q/182349 |website=Math Overflow|accessdate=2 October 2014}}</ref>
* '''Union and intersection''': The [[Union (set theory)|union]] or [[Intersection (set theory)|intersection]] of two star domains is not necessarily a star domain.
* '''Balance''': Given <math>W \subseteq X,</math> the set <math>\bigcap_{|u|=1} u W</math> (where <math>u</math> ranges over all [[unit length]] scalars) is a [[balanced set]] whenever <math>W</math> is a star shaped at the origin (meaning that <math>0 \in W</math> and <math>r w \in W</math> for all <math>0 \leq r \leq 1</math> and <math>w \in W</math>).
* '''Diffeomorphism''': A non-empty open star domain <math>S</math> in <math>\R^n</math> is [[Diffeomorphism|diffeomorphic]] to <math>\R^n.</math>
* '''Binary operators:''' If <math>A</math> and <math>B</math> are star domains, then so is the Cartesian product <math>A\times B</math>, and the sum <math>A + B</math>.<ref name=":0" />
* '''Linear transformations''': If <math>A</math> is a star domain, then so is every linear transformation of <math>A</math>.<ref name=":0" />

==See also==

* {{annotated link|Absolutely convex set}}
* {{annotated link|Absorbing set}}
* {{annotated link|Art gallery problem}}
* {{annotated link|Balanced set}}
* {{annotated link|Bounded set (topological vector space)}}
* {{annotated link|Convex set}}
* {{annotated link|Minkowski functional}}
* {{annotated link|Radial set}}
* {{annotated link|Star polygon}}
* {{annotated link|Symmetric set}}
* [[Star-shaped preferences]]

==References==

{{reflist}}
{{reflist|group=note}}

* 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.&nbsp;386, {{mr|0227724}}, {{JSTOR|2313423}}
* {{Rudin Walter Functional Analysis|edition=2}} <!--{{sfn|Rudin|1991|p=}}-->
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!--{{sfn|Schaefer|Wolff|1999|p=}}-->
* {{Schechter Handbook of Analysis and Its Foundations}} <!--{{sfn|Schechter|1996|p=}}-->

==External links==
{{commons category|Star-shaped sets}}

* {{mathworld|urlname=StarConvex|title=Star convex|author=Humphreys, Alexis}}

{{Functional analysis}}
{{Topological vector spaces}}
{{Convex analysis and variational analysis}}

[[Category:Convex analysis]]
[[Category:Euclidean geometry]]
[[Category:Functional analysis]]
[[Category:Linear algebra]]