star domain

For $V$ a vector space, a **star domain** about the origin is an inhabited subset $U \subset V$ such that with $v \in U$ and $s \in [0,1]$ also $s v \in U$.

More generally, for $X$ a real affine space, a **star domain** about a point $x\in X$ is an inhabited subset $U \subset X$ such that with $y \in X$, the straight line segment connecting $x$ with $y$ in $X$ is also contained in $U$.

These definitions can be modified in various obvious ways. For example, a **star shaped neighbourhood** of a point $x$ in an affine space $X$ is an open neighbourhood $U \subset X$ of $x$ that is a star domain about $x$. Or, a subset is a star domain if it is a star domain about one of its points.

A useful special case pertains to a simplicial complex $K$, where if $v$ is a vertex of $K$, then the **open star** of $v$ is the union of the interiors in ${|K|}$ of all the simplices containing $v$. Open stars of vertices provide a good open cover of a simplicial complex.

A convex set is the same as a set that is a star domain about each of its points.

Revised on September 21, 2015 12:20:02
by Todd Trimble
(67.81.95.215)