A bi-Lipschitz map 
between metric spaces is a map
for which constants 
satisfy
 |
for all 
. It is injective, and
both 
and its inverse on its image
are Lipschitz functions. A surjective
bi-Lipschitz map is also called a Lipschitz isomorphism of the metric
spaces. For Banach spaces, no linearity is included
in this terminology. When 
has at least two points, the metric embedding distortion is the product
of the Lipschitz constants of 
and its inverse on the
image.
