Star domain

In geometry, a set ${\displaystyle S}$ in the Euclidean space ${\displaystyle \mathbb {R} ^{n}}$ is called a star domain (or star-convex set, star-shaped set¹ or radially convex set) if there exists an ${\displaystyle s_{0}\in S}$ such that for all ${\displaystyle s\in S,}$ the line segment from ${\displaystyle s_{0}}$ to ${\displaystyle s}$ lies in ${\displaystyle S.}$ This definition is immediately generalizable to any real, or complex, vector space.

Intuitively, if one thinks of ${\displaystyle S}$ as a region surrounded by a wall, ${\displaystyle S}$ is a star domain if one can find a vantage point ${\displaystyle s_{0}}$ in ${\displaystyle S}$ from which any point ${\displaystyle s}$ in ${\displaystyle S}$ is within line-of-sight. A similar, but distinct, concept is that of a radial set.

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

A subset ${\displaystyle S}$ of a vector space ${\displaystyle X}$ is said to be star-shaped at ${\displaystyle s_{0}\in S}$ if for every ${\displaystyle s\in S,}$ the closed interval ${\displaystyle \left[s_{0},s\right]\subseteq S.}$ A set ${\displaystyle S}$ is star shaped and is called a star domain if there exists some point ${\displaystyle s_{0}\in S}$ such that ${\displaystyle S}$ is star-shaped at ${\displaystyle s_{0}.}$

A set that is star-shaped at the origin is sometimes called a star set.² Such sets are closely related to Minkowski functionals.

• Any line or plane in ${\displaystyle \mathbb {R} ^{n}}$ is a star domain.
• A line or a plane with a single point removed is not a star domain.
• If ${\displaystyle A}$ is a set in ${\displaystyle \mathbb {R} ^{n},}$ the set ${\displaystyle B=\{ta:a\in A,t\in [0,1]\}}$ obtained by connecting all points in ${\displaystyle A}$ 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 ${\displaystyle r<1,}$ the star domain can be dilated by a ratio ${\displaystyle r}$ such that the dilated star domain is contained in the original star domain.³
• Union and intersection: The union or intersection of two star domains is not necessarily a star domain.
• Balance: Given ${\displaystyle W\subseteq X,}$ the set ${\displaystyle \bigcap _{|u|=1}uW}$ (where ${\displaystyle u}$ ranges over all unit length scalars) is a balanced set whenever ${\displaystyle W}$ is a star shaped at the origin (meaning that ${\displaystyle 0\in W}$ and ${\displaystyle rw\in W}$ for all ${\displaystyle 0\leq r\leq 1}$ and ${\displaystyle w\in W}$).
• Diffeomorphism: A non-empty open star domain ${\displaystyle S}$ in ${\displaystyle \mathbb {R} ^{n}}$ is diffeomorphic to ${\displaystyle \mathbb {R} ^{n}.}$
• Binary operators: If ${\displaystyle A}$ and ${\displaystyle B}$ are star domains, then so is the Cartesian product ${\displaystyle A\times B}$, and the sum ${\displaystyle A+B}$.¹
• Linear transformations: If ${\displaystyle A}$ is a star domain, then so is every linear transformation of ${\displaystyle A}$.¹