A length metric, also called a path metric, is a metric 
on a set 
for which 
equals the infimum of the
lengths of paths joining 
to 
,
with lengths measured using 
(Birtea et al. 2008, p. 466; Lang 2013, p. 2).
For a path ![gamma:[0,1]->X](/images/equations/LengthMetric/Inline7.svg)
, its length with respect to 
is
 |
where the supremum ranges over all integers 
and all choices of 
satisfying the indicated inequalities. A metric
space equipped with a length metric is called a length
space.
