Definition A subset $U$ of a real (or possibly complex) vector space is called star-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 words, a region $U$ is star-shaped if there is a point $p\in U$ such that $U$ can be ``collapsed'' or ``contracted'' onto $p$

Examples

1. In $\sR^n$ any vector subspace is star-shaped. Also, the unit cube and unit 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.