Star domain

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

In geometry, a set S in the Euclidean space ${\displaystyle \mathbb {R} ^{n}}$ is called a star domain (or star-convex set, star-shaped set 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 s0 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 s0 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.

Examples

• 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.
• 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 ${\displaystyle r<1,}$ the star domain can be dilated by a ratio ${\displaystyle r}$ such that the dilated star domain is contained in the original star domain.[1]
• The union and intersection of two star domains is not necessarily a star domain.
• A non-empty open star domain ${\displaystyle S}$ in ${\displaystyle \mathbb {R} ^{n}}$ is diffeomorphic to ${\displaystyle \mathbb {R} ^{n}.}$
• 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 ${\displaystyle w\in W}$).

References

1. ^ Drummond-Cole, Gabriel C. "What polygons can be shrinked into themselves?". Math Overflow. Retrieved 2 October 2014.