Definition:Disk

Definition
Consider the Euclidean space $\left({\R^n, d}\right)$, where $d$ is the Euclidean metric.

An open $n$ dimensional disk (or ball) is defined as:


 * $\mathbb D^n = \left\{{x \in \R^n : d \left({x, y}\right) < r}\right\}$

where $y \in \R^n$ is called the center and $r \in \R_+$ is called the radius.

A closed $n$-disk is defined as:


 * $\mathbb D^n = \left\{{x \in \R^n : d \left({x, y}\right) \le r }\right\}$

The boundary of $\mathbb D^n$ is denoted $\partial \mathbb D^n$, and is $\mathbb S^{n-1}$, the $(n-1)$-sphere.

Note
The open disc of radius $r$ is a particular instance of an $r$-neighborhood in $\left({\R^n, d}\right)$.