Editing Metric space (section)

=== Miscellaneous examples === <!-- the goal is to remove these or incorporate them under the various subheadings -->
* Given a metric space {{math|(''X'', ''d'')}} and an increasing [[concave function]] <math>f \colon [0,\infty) \to [0,\infty)</math> such that {{math|''f''(''t'') {{=}} 0}} if and only if {{math|''t'' {{=}} 0}}, then <math>d_f(x,y)=f(d(x,y))</math> is also a metric on {{mvar|X}}.  If {{math|''f''(''t'') {{=}} ''t''<sup>α</sup>}} for some real number {{math|α < 1}}, such a metric is known as a '''snowflake''' of {{mvar|d}}.<ref>{{cite conference |last1=Gottlieb |first1=Lee-Ad |last2=Solomon |first2=Shay |title=Light spanners for snowflake metrics |conference=SOCG '14: Proceedings of the thirtieth annual symposium on Computational geometry |date=8 June 2014 |pages=387–395 |doi=10.1145/2582112.2582140|arxiv=1401.5014 }}</ref>
* The [[tight span]] of a metric space is another metric space which can be thought of as an abstract version of the [[convex hull]].
* The ''knight's move metric'', the minimal number of knight's moves to reach one point in <math>\mathbb{Z}^2</math> from another, is a metric on <math>\mathbb{Z}^2</math>.
* {{anchor|SNCF}}The [[British Rail]] metric (also called the "post office metric" or the "[[French railway metrics|French railway metric]]") on a [[normed vector space]] is given by <math>d(x,y) = \lVert x \rVert + \lVert y \rVert</math> for distinct points <math>x</math> and <math>y</math>, and <math>d(x,x) = 0</math>. More generally <math>\lVert \cdot \rVert</math> can be replaced with a function <math>f</math> taking an arbitrary set <math>S</math> to non-negative reals and taking the value <math>0</math> at most once:  then the metric is defined on <math>S</math> by <math>d(x,y) = f(x) + f(y)</math> for distinct points <math>x</math> and <math>y</math>, and {{nowrap|<math>d(x,x) = 0</math>.}}  The name alludes to the tendency of railway journeys to proceed via London (or Paris) irrespective of their final destination.<!-- source? -->
* The [[Robinson–Foulds metric]] used for calculating the distances between [[Phylogenetic tree]]s in [[Phylogenetics]]<ref>{{Cite journal|last1=Robinson|first1=D.F.|last2=Foulds|first2=L.R.|date=February 1981|title=Comparison of phylogenetic trees|url=https://linkinghub.elsevier.com/retrieve/pii/0025556481900432|journal=Mathematical Biosciences|language=en|volume=53|issue=1–2|pages=131–147|doi=10.1016/0025-5564(81)90043-2|s2cid=121156920 |url-access=subscription}}</ref>