Uniform rectifiability is a quantitative form of rectifiability requiring substantial images of Lipschitz functions at every location
and scale. An 
-dimensional
Ahlfors-David regular measure 
on 
is uniformly 
-rectifiable if there are constants 
and 
such that, for every 
and 
, some Lipschitz
function 
with Lipschitz constant at most 
satisfies
 |
Here the balls are open, 
is the support, and 
denotes diameter. For example, Hausdorff
measure on an 
-dimensional affine space has
this property using an isometric parametrization. The
uniform constants strengthen the covering condition
in the definition of a rectifiable set.
