Definition:Ball

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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


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