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I've found some interesting metrics other than the euclidean metric, such as the taxi cab metric and the British rail metric. Are there any other interesting metrics out there? Please give some sort of visualization if possible.

Mark S
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  • there's the discrete metric (if $x = y$ then $d(x,y) = 0$; otherwise, $d(x,y) = 1$; maybe not so interesting) – J. W. Tanner Jan 06 '20 at 02:43
  • @J. W. Tanner would a ball of radius 1 with (0,0) in $\mathbb{R}^2$ be all points in $\mathbb{R}^2$ except (0,0)? – Mark S Jan 06 '20 at 02:50
  • in the discrete metric, the open ball of radius $1$ about a point would be only that point, and the closed ball would be the whole space – J. W. Tanner Jan 06 '20 at 02:52
  • more interesting: a $p$-adic metric space with $p$ prime can be formed on the set of rational numbers with $d(x,x)=0$ and $d(x,y)=p^{-\nu_p(x)+\nu_p(y)}$ if $x\ne y$ where $\nu_p(n)=\max{v\in\mathbb N:p^v $ divides $n} $ if $n\ne0$ and $\infty$ if $n=0$ – J. W. Tanner Jan 06 '20 at 02:56
  • @J. W. Tanner, what does p-adic mean and what would a ball look like in this metric space? – Mark S Jan 06 '20 at 03:04
  • integers are close if their difference is divisible by a high power of $p$; in my comment above, I should have said $d(x,y)=p^{-\nu_p(x-y)}$ and $\nu_p(r/s)=\nu_p(r)-\nu_p(s)$, where $r$ and $s$ are integers – J. W. Tanner Jan 06 '20 at 03:10
  • Ostrowski's theorem states that every absolute value on the rational numbers is equivalent to the trivial absolute value, the usual real absolute value, or a p-adic absolute value – J. W. Tanner Jan 06 '20 at 03:20
  • I can't really visualize what a ball would look like in this metric space. – Mark S Jan 06 '20 at 03:26
  • Any convex set symmetric about all axes induces a metric by scaling copies around it's center. – CyclotomicField Jan 06 '20 at 03:28
  • in the $p$-adic metric, the closed ball around $0$ of radius $1/p$ would be rational numbers with positive powers of $p$ in their numerators – J. W. Tanner Jan 06 '20 at 03:39

1 Answers1

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  1. The river metric on $\Bbb R^2: \; d((x,y),(x,y'))=|y-y'|$ but if $x\ne x'$ then $d((x,y),(x',y'))=|x-x'|+|y|+|y'|.$ The idea is that if $x\ne x'$ then $\{x\}\times \Bbb R$ is separated from $\{x'\}\times \Bbb R$ by mountains, so to travel from $(x,y)$ to $(x',y')$ you must travel to , and along the river $\Bbb R\times \{0\}.$

  2. A metric on $\Bbb R\cup \{p\}$ with $p\not \in \Bbb R:\; d(x,x')=\frac {|x-x'|}{1+|x-x'|}$ when $x,x'\in \Bbb R,$ and $d(p,x)=2-d(0,x)$ when $x\in \Bbb R.$ This gives a locally-compact complete metric space with a closed subset $\Bbb R$ such that $\inf \{d(p,x):x\in \Bbb R\}=1,$ but no $x\in \Bbb R$ satisfies $d(p,x)=1.$

  3. For $1\le p\in \Bbb R$ the $l_p$ metric $d_p((x,y),(x',y'))=(|x-x'|^p+|y-y'|^p)^{1/p}$ generates the standard topology (i.e. the topology from $p=2$) on $\Bbb R^2.$ As $p\to \infty,$ the unit sphere $\{(x,y): d_p((x,y),(0,0))=1\}$ approaches $\{(x,y):\max (|x|,|y|)=1\},$ which is the unit sphere of the $l_{\infty}$ metric $d_{\infty}((x,y),(x',y'))=\max (|x-x'|,|y-y'|),$ which also generates the same topology. This generalizes to $\Bbb R^n$ for $2\le n \in \Bbb N.$

  4. If $(X,d)$ and $(Y,e)$ are metric spaces and if $f:X\to Y$ is continuous, then on $X,$ the metric $d'(x,x')=d(x,x')+e(f(x),f(x'))$ on $X$ is equivalent to $d.$

    That is, $d$ and $d'$ generate the same topology on $X$.

    Particularly when $Y=\Bbb R$ and $e(y,y')=|y-y'|,$ this is useful.

    For example if $(X,d)$ is a non-compact metric space, we show that $X$ has an infinite closed discrete subspace $S$ and show there is a continuous $f:X\to \Bbb R$ such that $\{f(x):x\in S\}$ is unbounded in $\Bbb R,$ in order to prove that the $d$-topology on $X$ can be generated by an unbounded metric, namely by $d'(x,x')=d(x,x')+|f(x)-f(x')|.$

  5. For any metric space $(X,d)$ the metrics $d_1(x,x')=\min (1,d(x,x'))$ and $d_2(x,x')=\frac {d(x,x')}{1+d(x,x')}$ are equivalent to $d.$ This illustrates that "bounded subset of $X$" is not generally a topological property, but (unless $X$ is compact) depends on the metric, not on the topology that the metric generates.

DanielWainfleet
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