Star domain

Given two points $x$ and $y$ in a vector space $X$ (such as Euclidean space $\backslash R^n$), the convex hull of $\backslash \{x,\; y\backslash \}$ is called the {{em|closed interval with endpoints $x$ and $y$}} and it is denoted by

$$\backslash left[x,\; y\backslash right]\; ~:=~\; \backslash left\backslash \{t\; y\; +\; (1\; -\; t)\; x\; :\; 0\; \backslash leq\; t\; \backslash leq\; 1\backslash right\backslash \}\; ~=~\; x\; +\; (y\; -\; x)\; [0,\; 1],$$

where $z\; [0,\; 1]\; :=\; \backslash \{z\; t\; :\; 0\; \backslash leq\; t\; \backslash leq\; 1\backslash \}$ for every vector $z.$

A subset $S$ of a vector space $X$ is said to be {{em|star-shaped at}} $s\_0\; \backslash in\; S$ if for every $s\; \backslash in\; S,$ the closed interval

$\backslash left[s\_0,\; s\backslash right]\; \backslash subseteq\; S.$

A set $S$ is {{em|star shaped}} and is called a {{em|star domain}} if there exists some point $s\_0\; \backslash in\; S$ such that $S$ is star-shaped at $s\_0.$

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 functionals.

• Any line or plane in $\backslash R^n$ 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 $\backslash R^n,$ the set $B\; =\; \backslash \{t\; a\; :\; a\; \backslash in\; A,\; t\; \backslash in\; [0,\; 1]\backslash \}$ obtained by connecting all points in $A$ 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.

• Convexity: 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 each point in that set.
• Closure and interior: The closure of a star domain is a star domain, but the interior of a star domain is not necessarily a star domain.
• Contraction: Every star domain is a contractible set, via a 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 the star domain can be dilated by a ratio $r$ such that the dilated star domain is contained in the original star domain.{{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}}
• Union and intersection: The union or intersection of two star domains is not necessarily a star domain.
• Balance: Given $W\; \backslash subseteq\; X,$ the set $\backslash bigcap\_\{|u|=1\}\; u\; W$ (where $u$ ranges over all unit length scalars) is a balanced set whenever $W$ is a star shaped at the origin (meaning that $0\; \backslash in\; W$ and $r\; w\; \backslash in\; W$ for all $0\; \backslash leq\; r\; \backslash leq\; 1$ and $w\; \backslash in\; W$).
• Diffeomorphism: A non-empty open star domain $S$ in $\backslash R^n$ is diffeomorphic to $\backslash R^n.$
• Binary operators: If $A$ and $B$ are star domains, then so is the Cartesian product $A\backslash times\; B$, and the sum $A\; +\; B$.
• Linear transformations: If $A$ is a star domain, then so is every linear transformation of $A$.

• {{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

{{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. 386, {{mr|0227724}}, {{JSTOR|2313423}}
• {{Rudin Walter Functional Analysis|edition=2}}
• {{Schaefer Wolff Topological Vector Spaces|edition=2}}
• {{Schechter Handbook of Analysis and Its Foundations}}

{{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