star-shaped region

DefinitionA subset $U$ of a real (or possibly complex) vector space is calledstar-shaped if there is a point $p\in U$ such that the line segment$\overline{pq}$ is contained in $U$ for all $q\in U$. (Here, $\overline{pq}=\{tp+(1-t)q|t\in [0,1]\}$.) We then say that $U$ is star-shaped with respect to $p$.

In other , a region $U$ is star-shaped if there is a point $p\in U$ such that $U$ can be “collapsed” or “contracted” $p$.

0.0.1 Examples

1. In ${ℝ}^n$, any vector subspace is star-shaped. Also, the unit cube andunit ball are star-shaped, but the unit sphere is not.

2. A subset $U$ of a vector space is star-shaped with respect to all of its points if and only if $U$ is convex.