# star-shaped region

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 , 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. 1.

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

2. 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.

Title star-shaped region StarshapedRegion 2013-03-22 13:34:13 2013-03-22 13:34:13 matte (1858) matte (1858) 10 matte (1858) Definition msc 52A30 msc 32F99 star-shaped