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

A subset $$S$$ of a vector space $$X$$ is said to be $$s_0 \in S$$ if for every $$s \in S,$$ the closed interval $$\left[s_0, s\right] \subseteq S.$$ A set $$S$$ is  and is called a  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. 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 $$B = \{t a : a \in A, t \in [0, 1]\}$$ 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 each 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 $$\R^n$$ is diffeomorphic to $$\R^n.$$
 * Given $$W \subseteq X,$$ the set $$\bigcap_{|u|=1} u W$$ (where $$u$$ ranges over all unit length scalars) is a balanced set whenever $$W$$ is a star shaped at the origin (meaning that $$0 \in W$$ and $$r w \in W$$ for all $$0 \leq r \leq 1$$ and $$w \in W$$).