Star domain

"Star-shaped" redirects here. For the Blur documentary, see Starshaped.

File:Star domain.svg

A star domain (equivalently, a star-convex or star-shaped set) is not necessarily convex in the ordinary sense.

File:Not-star-shaped.svg

An annulus is not a star domain.

In geometry, a set $S$ in the Euclidean space ${ℝ}^n$ is called a star domain (or star-convex set, star-shaped set or radially convex set) if there exists an $s_0\in S$ such that for all $s\in S,$ the line segment from $s_0$ to $s$ lies in $S.$ This definition is immediately generalizable to any real, or complex, 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.

Definition

Given two points $x$ and $y$ in a vector space $X$ (such as Euclidean space ${ℝ}^n$), the convex hull of $\{x,y\}$ is called the closed interval with endpoints $x$ and $y$ and it is denoted by $$[x,y]:=\{ty+(1-t)x:0\le t\le 1\}=x+(y-x)[0,1],$$ where $z[0,1]:=\{zt:0\le t\le 1\}$ for every vector $z.$

A subset $S$ of a vector space $X$ is said to be star-shaped at $s_0\in S$ if for every $s\in S,$ the closed interval $[s_0,s]\subseteq S.$ A set $S$ is star shaped and is called a star domain if there exists some point $s_0\in S$ such that $S$ is star-shaped at $s_0.$

A set that is star-shaped at the origin is sometimes called a star set. Such sets are closely related to Minkowski functionals.

Examples

• Any line or plane in ${ℝ}^n$ 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 ${ℝ}^n,$ the set $B=\{ta:a\in A,t\in [0,1]\}$ 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.

Properties

• 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 $r<1,$ 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 $W\subseteq X,$ the set $\bigcap _{|u|=1}uW$ (where $u$ ranges over all unit length scalars) is a balanced set whenever $W$ is a star shaped at the origin (meaning that $0\in W$ and $rw\in W$ for all $0\le r\le 1$ and $w\in W$).
• Diffeomorphism: A non-empty open star domain $S$ in ${ℝ}^n$ is diffeomorphic to ${ℝ}^n.$
• Binary operators: If $A$ and $B$ are star domains, then so is the Cartesian product $A\times B$, and the sum $A+B$.
• Linear transformations: If $A$ is a star domain, then so is every linear transformation of $A$.

See also

• Absolutely convex set – Convex and balanced set
• Absorbing set – Set that can be "inflated" to reach any point
• Art gallery problem – Mathematical problem
• Balanced set – Construct in functional analysis
• Bounded set (topological vector space) – Generalization of boundedness
• Convex set – In geometry, set whose intersection with every line is a single line segment
• Minkowski functional – Function made from a set
• Radial set – Topological set
• Star polygon – Regular non-convex polygon
• Symmetric set – Property of group subsets (mathematics)
• Star-shaped preferences

References

• 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, , JSTOR 2313423