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), 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 balls can be found here.
Sources
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): disk or disc
- 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