Definition:Closed Ball/Real Euclidean Space

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Definition

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

Let $\R^n$ denote the real Euclidean space of dimension $n$.

Let $\norm {\, \cdot \,}$ denote the Euclidean norm.

Let $a \in \R^n$.

Let $\epsilon \in \R_{>0}$ be a strictly positive real number.


The closed (Euclidean) ball of $\R^n$ of center $a$ and radius $\epsilon$ is the subset:

$\map { { {\overline B}_\epsilon}^n} a = \set {x \in \R^n : \norm {x - a} \le \epsilon}$

If the dimension $n$ has already been established, then it is commonplace to simplify the notation and present it as $\map { {\overline B}_\epsilon} a$.


Examples

Dimension $2$

Let $\map { { {\overline B}_\epsilon}^2} a$ be the closed Euclidean ball of center $a$ and radius $\epsilon$ in the Euclidean plane $\R^2$.

Then $\map { { {\overline B}_\epsilon}^2} a$ is of the form of a disc of radius $\epsilon$ and center $a$.


Dimension $1$

Let $\map { { {\overline B}_\epsilon}^1} a$ be the closed Euclidean ball of center $a$ and radius $\epsilon$ in the real number line $\R$.

Then $\map { { {\overline B}_\epsilon}^1} a$ is the closed real interval $\closedint {a - \epsilon} {a + \epsilon}$.


Also defined as

Some sources define a closed Euclidean ball with a radius equal to $1$ and a center set to the origin, that is:

${\overline B}^n := \set {x \in \R^n : \norm x \le 1}$

but it needs to be appreciated that this definition is needlessly limiting.


Also known as

Some sources refer to a closed Euclidean ball of $n$ dimensions merely as an $n$-ball, as they are not interested in contexts other than real Euclidean spaces.

However, it needs to be remembered that a ball has a more general definition in the context of a general metric space.


Variant Terminology for Ball

Instead of ball, some sources use the term disk (or disc in British English), and prefer to use $\Bbb D$ for $B$.

Some sources use disk (or disc) specifically to mean closed ball, and use open disk (or open disc) for open ball.

$\mathsf{Pr} \infty \mathsf{fWiki}$ prefers to reserve the term disk, if at all, for a disk in the complex plane, as there is an intuitive $2$-dimensional nuance to the word disk, while ball guides intuition down the path of $3$ dimensions.

The Concise Oxford Dictionary of Mathematics distinguishes between a disc, which is what it is in its context of a circle in the plane, and a disk, which is used as a synonym for an open or closed ball in a general metric space.

However, this is not how we roll at $\mathsf{Pr} \infty \mathsf{fWiki}$, where the aim is that open ball and closed ball are to be used consistently.


Also see

  • Results about closed Euclidean balls can be found here.


Sources

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