Definition:Disk

Euclidean disk
Let $n \ge 1 $ be a natural number.

Let $\struct {\R^n, d}$ 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:


 * $\map {\mathbb D^n} {a, r} = \set {x \in \R^n: \map d {x, a} < r}$

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


 * $\map {\overline {\mathbb D}^n} {a, r} = \set {x \in \R^n: \map d {x, a} \le r}$

The boundary of $\mathbb D^n$ is denoted $\partial \mathbb D^n$, and is $\mathbb S^{n - 1}$, the $\paren {n - 1}$-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 $\struct {\R^n, d}$.