StarShapedRegion 2003-04-13 13:15:00 2006-10-31 11:30:05 Definition star-shaped % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \newcommand{\sR}[0]{\mathbb{R}} \newcommand{\sC}[0]{\mathbb{C}} \newcommand{\sN}[0]{\mathbb{N}} \newcommand{\sZ}[0]{\mathbb{Z}} {\bf Definition} A subset $U$ of a real (or possibly complex) vector space is called \emph{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 \PMlinkescapetext{words}, a region $U$ is star-shaped if there is a point $p\in U$ such that $U$ can be ``collapsed'' or ``contracted'' \PMlinkescapetext{onto} $p$. \subsubsection{Examples} \begin{enumerate} \item 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. \item A subset $U$ of a vector space is star-shaped with respect to all of its points if and only if $U$ is convex. \end{enumerate}