An Ahlfors-David regular measure of dimension 
on 
,
for integers 
,
is a Radon measure 
for which a constant 
satisfies
 |
for every 
in the support 
of 
and every 
.
Here 
is the open ball of radius 
, and 
is the diameter
of 
. The definition is also abbreviated AD regularity.
For example, 
-dimensional Hausdorff measure
restricted to an 
-dimensional affine space in
is Ahlfors-David regular. With the normalization that agrees with Lebesgue
measure on 
,
its measure on each ball centered
in that affine space is 
, where 
is the volume of the unit
ball in 
(Mattila 2023, pp. 4-5). The two-sided growth condition
is an assumption in the David-Semmes problem.
