# Definition:Ball

## Definition

### Open Ball

Let $M = \struct {A, d}$ be a metric space or pseudometric space.

Let $a \in A$.

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

The **open $\epsilon$-ball of $a$ in $M$** is defined as:

- $\map {B_\epsilon} a := \set {x \in A: \map d {x, a} < \epsilon}$

If it is necessary to show the metric or pseudometric itself, then the notation $\map {B_\epsilon} {a; d}$ can be used.

### Closed Ball

Let $M = \struct {A, d}$ be a metric space.

Let $a \in A$.

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

The **closed $\epsilon$-ball of $a$ in $M$** is defined as:

- $\map { {B_\epsilon}^-} a := \set {x \in A: \map d {x, a} \le \epsilon}$

where $B^-$ recalls the notation of topological closure.

If it is necessary to show the metric itself, then the notation $\map { {B_\epsilon}^-} {a; d}$ can be used.

### Unit Ball

Let $V$ be a normed vector space with norm $\norm {\, \cdot \,}$.

The **closed unit ball** of $V$, denoted $\operatorname {ball} V$, is the set:

- $\set {v \in V: \norm v_V \mathop \le 1}$

## Also known as

Instead of **ball**, some sources use the term **disk** (or **disc]** in British English).

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
**balls**can be found**here**.

## Sources

- 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**disk**or**disc** - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**disc (disk)** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**disc (disk)** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**ball** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**disk**

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- 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**disc**