Definition:Disk

Higher dimensional disks
Let $n\geq 1 $ be a natural number.

Let $\left({\R^n, d}\right)$ be the $n$-dimensional Euclidean space, where $d$ is the Euclidean metric.

Let $a\in \R^n$.

Let $r>0$ be a real number.

Open disk
The open $n$-disk of center $a$ and radius $r$ is the set:


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

Closed disk
A closed $n$-disk of center $a$ and radius $r$ is the set:


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

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

Also see

 * Definition:Unit Disk
 * Definition:Polydisk
 * Definition:Ball
 * Definition:Sphere
 * Boundary of Disk is Sphere
 * The open disc of radius $r$ is a particular instance of an open $r$-ball in $\left({\R^n, d}\right)$.