Star domain

 ( Convex Analysis ) 

In geometry, a set $S$ in the Euclidean space $\backslash R^n$ is called a star domain (or star-convex set, star-shaped set or radially convex set) if there exists an $s\_0\; \backslash in\; S$ such that for all $s\; \backslash in\; S,$ the line segment from $s\_0$ to $s$ lies in $S.$ This definition is immediately generalizable to any Real number, or Complex number, vector space.

Intuitively, if one thinks of $S$ as a region surrounded by a wall, $S$ is a star domain if one can find a vantage point $s\_0$ in $S$ from which any point $s$ in $S$ is within line-of-sight. A similar, but distinct, concept is that of a radial set.

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Definition 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 and it is denoted by $$\backslash leftx,\; ~:=~\; \backslash left\backslash \{t\; y\; +\; (1\; -\; t)\; x\; :\; 0\; \backslash leq\; t\; \backslash leq\; 1\backslash right\backslash \}\; ~=~\; x\; +\; (y\; -\; x)\; 0,,$$ where $z\; 0,\; :=\; \backslash \{z\; t\; :\; 0\; \backslash leq\; t\; \backslash leq\; 1\backslash \}$ for every vector $z.$

A subset $S$ of a vector space $X$ is said to be $s\_0\; \backslash in\; S$ if for every $s\; \backslash in\; S,$ the closed interval $\backslash lefts\_0,\; \backslash subseteq\; S.$ A set $S$ is and is called a 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 . Such sets are closely related to Minkowski functionals.

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Examples

• 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 $B\; =\; \backslash \{t\; a\; :\; a\; \backslash in\; A,\; t\; \backslash in\; 0,\backslash \}$ 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.

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Properties

• Convexity: any Empty set 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.
• 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 Diffeomorphism 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 $A$.

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See also

• Star-shaped preferences

• Ian Stewart, David Tall, Complex Analysis. Cambridge University Press, 1983, ,
• C.R. Smith, A characterization of star-shaped sets, American Mathematical Monthly, Vol. 75, No. 4 (April 1968). p. 386, ,

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External links

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