standard n-ball

The standard n-ball is $B^n=\{x\in {ℝ}^n|\mathit{\hspace{1em}}||x||\le 1\}$, where ${||x||}^2={\sum }_{i=1}^{n}x_i{}^2$. The standard n-ball is also called the unit n-ball or the 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 n-cell.

A 1-cell is called an arc and a 2-cell is called a disk.

Title	standard n-ball
Canonical name	StandardNball
Date of creation	2013-03-22 16:52:49
Last modified on	2013-03-22 16:52:49
Owner	Mathprof (13753)
Last modified by	Mathprof (13753)
Numerical id	6
Author	Mathprof (13753)
Entry type	Definition
Classification	msc 55M99
Synonym	unit n-ball
Synonym	unit ball
Synonym	standard $n$-ball
Defines	n-cell