Hausdorff dimension
The new approach of the notion of dimension can be defined rigorously. Given two real numbers
and
, one can overlap the studied set
by
sphere of diameter
,
all lower that
(see the figure), then one calculate the limit :

The limit when
tends to 0 of the minorant, corresponds to all spheres of the sum of
power of

One can show there exist one value of
such as:

has a finite value for
.
is the Hausdorff dimension of the set
.