Star domain

In mathematics, a set ${\displaystyle S}$ 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 ${\displaystyle s_{0}}$ to ${\displaystyle s}$ lies in ${\displaystyle S.}$ This definition is immediately generalizable to any real or complex vector space.

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.

Intuitively, if one thinks of ${\displaystyle S}$ as of 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.