StandardNBall 2007-03-30 16:41:37 2007-05-12 16:23:24 Definition n-cell % 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 The \emph{standard n-ball} is $B^n = \{x \in \mathbb{R}^n | \quad ||x|| \le 1 \}$, where $||x||^2 = \sum_{i=1}^n {x_i}^2 $. The standard n-ball is also called the \emph{unit n-ball} or the \emph{n-disk}. The notation in the literature for the standard n-ball is not uniform. Sometimes it is denoted by $V^n$ , $D^n$ ,$E^n$ or ${\overline{E}}^n$. The boundary of the standard n-ball is $S^{n-1}$, the n-1-sphere. A space that is homeomorphic to the standard n-ball is called an \emph{n-cell}. A 1-cell is called an arc and a 2-cell is called a disk.