A doubling metric space is a metric space 
for which some finite constant 
has the following property. Every
ball of radius 
can be covered by at most 
balls of radius 
, with centers in 
. The least such integer
is its doubling constant. This gives a bound on local
covering complexity independent of scale. For example, the real
line with its usual metric has doubling constant
2 using closed balls.
The Lang-Plaut problem asks whether every doubling subset of a Hilbert space can be represented in a finite-dimensional Euclidean space with finite metric embedding distortion.
