# Star domain

In geometry, a set $S$ in the Euclidean space $\mathbb {R} ^{n}$ 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 $\mathbb {R} ^{n}$ ), the convex hull of $\{x,y\}$ is called the closed interval with endpoints $x$ and $y$ and it is denoted by

$\left[x,y\right]~:=~\left\{tx+(1-t)y:0\leq t\leq 1\right\}~=~x+(y-x)[0,1],$ where $z[0,1]:=\{zt:0\leq t\leq 1\}$ 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 $\left[s_{0},s\right]\subseteq S.$ 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 closed related to Minkowski functionals.

## Examples

• Any line or plane in $\mathbb {R} ^{n}$ 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 $\mathbb {R} ^{n},$ 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.
• 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.
• A star-shaped polygon is a star domain whose boundary is a sequence of connected line segments.

## Properties

• The closure of a star domain is a star domain, but the interior of a star domain is not necessarily a star domain.
• Every star domain is a contractible set, via a straight-line homotopy. In particular, any star domain is a simply connected set.
• 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.
• The union and intersection of two star domains is not necessarily a star domain.
• A non-empty open star domain $S$ in $\mathbb {R} ^{n}$ is diffeomorphic to $\mathbb {R} ^{n}.$ • 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 $0\leq r\leq 1$ and $w\in W$ ).