# Dimension function

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In [mathematics](/source/mathematics), the notion of an ('''exact''') '''dimension function''' (also known as a '''gauge function''') is a tool in the study of [fractal](/source/fractal)s and other subsets of [metric space](/source/metric_space)s. Dimension functions are a generalisation of the simple "[diameter](/source/diameter) to the [dimension](/source/dimension)" [power law](/source/power_law) used in the construction of ''s''-dimensional [Hausdorff measure](/source/Hausdorff_measure).

==Motivation: ''s''-dimensional Hausdorff measure==

{{main|Hausdorff dimension}}

Consider a metric space (''X'',&nbsp;''d'') and a [subset](/source/subset) ''E'' of ''X''. Given a number ''s''&nbsp;≥&nbsp;0, the ''s''-dimensional '''Hausdorff measure''' of ''E'', denoted ''μ''<sup>''s''</sup>(''E''), is defined by

:<math>\mu^{s} (E) = \lim_{\delta \to  0} \mu_{\delta}^{s} (E),</math>

where

:<math>\mu_{\delta}^{s} (E) = \inf \left\{ \left. \sum_{i = 1}^{\infty} \mathrm{diam} (C_{i})^{s} \right| \mathrm{diam} (C_{i}) \leq \delta, \bigcup_{i = 1}^{\infty} C_{i} \supseteq E \right\}.</math>

''μ''<sub>''δ''</sub><sup>''s''</sup>(''E'') can be thought of as an approximation to the "true" ''s''-dimensional area/volume of ''E'' given by calculating the minimal ''s''-dimensional area/volume of a covering of ''E'' by sets of diameter at most ''δ''.

As a function of increasing ''s'', ''μ''<sup>''s''</sup>(''E'') is non-increasing. In fact, for all values of ''s'', except possibly one, ''H''<sup>''s''</sup>(''E'') is either 0 or +∞; this exceptional value is called the '''Hausdorff dimension''' of ''E'', here denoted dim<sub>H</sub>(''E''). Intuitively speaking, ''μ''<sup>''s''</sup>(''E'')&nbsp;=&nbsp;+∞ for ''s''&nbsp;&lt;&nbsp;dim<sub>H</sub>(''E'') for the same reason as the 1-dimensional linear [length](/source/length) of a 2-dimensional [disc](/source/Disk_(mathematics)) in the [Euclidean plane](/source/Euclidean_plane) is +∞; likewise, ''μ''<sup>''s''</sup>(''E'')&nbsp;=&nbsp;0 for ''s''&nbsp;&gt;&nbsp;dim<sub>H</sub>(''E'') for the same reason as the 3-dimensional [volume](/source/volume) of a disc in the Euclidean plane is zero.

The idea of a dimension function is to use different functions of diameter than just diam(''C'')<sup>''s''</sup> for some ''s'', and to look for the same property of the Hausdorff measure being finite and non-zero.

==Definition==

Let (''X'',&nbsp;''d'') be a metric space and ''E''&nbsp;⊆&nbsp;''X''. Let ''h''&nbsp;:&nbsp;[0,&nbsp;+∞)&nbsp;→&nbsp;[0,&nbsp;+∞] be a function. Define ''μ''<sup>''h''</sup>(''E'') by

:<math>\mu^{h} (E) = \lim_{\delta \to  0} \mu_{\delta}^{h} (E),</math>

where

:<math>\mu_{\delta}^{h} (E) = \inf \left\{ \left. \sum_{i = 1}^{\infty} h \left( \mathrm{diam} (C_{i}) \right) \right| \mathrm{diam} (C_{i}) \leq \delta, \bigcup_{i = 1}^{\infty} C_{i} \supseteq E \right\}.</math>

Then ''h'' is called an ('''exact''') '''dimension function''' (or '''gauge function''') for ''E'' if ''μ''<sup>''h''</sup>(''E'') is finite and strictly positive. There are many conventions as to the properties that ''h'' should have: Rogers (1998), for example, requires that ''h'' should be [monotonically increasing](/source/monotone_function) for ''t''&nbsp;≥&nbsp;0, strictly positive for ''t''&nbsp;&gt;&nbsp;0, and [continuous](/source/continuous_function) on the right for all ''t''&nbsp;≥&nbsp;0.

===Packing dimension===

[Packing dimension](/source/Packing_dimension) is constructed in a very similar way to Hausdorff dimension, except that one "packs" ''E'' from inside with [pairwise disjoint](/source/disjoint_sets) balls of diameter at most ''δ''.  Just as before, one can consider functions ''h''&nbsp;:&nbsp;[0,&nbsp;+∞)&nbsp;→&nbsp;[0,&nbsp;+∞] more general than ''h''(''δ'')&nbsp;=&nbsp;''δ''<sup>''s''</sup> and call ''h'' an exact dimension function for ''E'' if the ''h''-packing measure of ''E'' is finite and strictly positive.

==Example==

[Almost surely](/source/Almost_surely), a sample path ''X'' of [Brownian motion](/source/Brownian_motion) in the Euclidean plane has Hausdorff dimension equal to 2, but the 2-dimensional Hausdorff measure ''μ''<sup>2</sup>(''X'') is zero. The exact dimension function ''h'' is given by the [logarithm](/source/logarithm)ic correction

:<math>h(r) = r^{2} \cdot \log \frac1{r} \cdot \log \log \log \frac1{r}.</math>

I.e., with probability one, 0&nbsp;&lt;&nbsp;''μ''<sup>''h''</sup>(''X'')&nbsp;&lt;&nbsp;+∞ for a Brownian path ''X'' in '''R'''<sup>2</sup>.  For Brownian motion in Euclidean ''n''-space '''R'''<sup>''n''</sup> with ''n''&nbsp;&ge;&nbsp;3, the exact dimension function is

:<math>h(r) = r^{2} \cdot \log \log \frac1r.</math>

==References==
{{reflist}}
* {{cite journal
|   author = Olsen, L.
|    title = The exact Hausdorff dimension functions of some Cantor sets
|  journal = Nonlinearity
|   volume = 16
|     year = 2003
|    issue = 3
|    pages = 963&ndash;970
|    doi = 10.1088/0951-7715/16/3/309
}}
* {{cite book
|    author = Rogers, C. A.
|     title = Hausdorff measures
|   edition = Third
|    series = Cambridge Mathematical Library
| publisher = Cambridge University Press
|  location = Cambridge
|      year = 1998
|     pages = xxx+195
|        isbn = 0-521-62491-6
}}

Category:Dimension theory
Category:Fractals
Category:Metric geometry

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Adapted from the Wikipedia article [Dimension function](https://en.wikipedia.org/wiki/Dimension_function) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Dimension_function?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.