Definition:Disk

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 $$\left\{{\R^n, d}\right\}$$.