A capacity dimension is a dimension also called the fractal dimension, Hausdorff
dimension, and Hausdorff-Besicovitch dimension in which nonintegral values are
permitted. Objects whose capacity dimension is different from their Lebesgue
covering dimension are called fractals. The capacity
dimension of a compact metric space 
is a real number 
such that if 
denotes the minimum number of open sets of diameter
less than or equal to 
, then 
is proportional to 
as 
. Explicitly,
 |
(if the limit exists), where 
is the number of elements forming a finite cover
of the relevant metric space and 
is a bound on the diameter of the sets involved (informally,
is the size of each element used to cover the set, which is taken to approach 0).
If each element of a fractal is equally likely to be
visited, then 
,
where 
is the information dimension.
The capacity dimension satisfies
 |
where 
is the correlation dimension (correcting
the typo in Baker and Gollub 1996).
