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

2. 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
Canonical name	StarshapedRegion
Date of creation	2013-03-22 13:34:13
Last modified on	2013-03-22 13:34:13
Owner	matte (1858)
Last modified by	matte (1858)
Numerical id	10
Author	matte (1858)
Entry type	Definition
Classification	msc 52A30
Classification	msc 32F99
Defines	star-shaped