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  "slug": "dimension-function",
  "title": "Dimension function",
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  "lead": "In mathematics, the notion of an (exact) dimension function (also known as a gauge function) is a tool in the study of fractals and other subsets of metric spaces. Dimension functions are a generalisation of the simple \"diameter to the dimension\" power law used in the construction of s-dimensional Hausdorff measure.",
  "body": "In mathematics, the notion of an (exact) dimension function (also known as a gauge function) is a tool in the study of fractals and other subsets of metric spaces. Dimension functions are a generalisation of the simple \"diameter to the dimension\" power law used in the construction of s-dimensional Hausdorff measure. ==Motivation: s-dimensional Hausdorff measure== Consider a metric space (X, d) and a subset E of X. Given a number s ≥ 0, the s-dimensional Hausdorff measure of E, denoted μ s (E), is defined by \\mu^{s} (E) = \\lim_{\\delta \\to 0} \\mu_{\\delta}^{s} (E), where \\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\\}. μ δ s (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 s (E) is non-increasing. In fact, for all values of s, except possibly one, H s (E) is either 0 or +∞; this exceptional value is called the Hausdorff dimension of E, here denoted dim H (E). Intuitively speaking, μ s (E) = +∞ for s < dim H (E) for the same reason as the 1-dimensional linear length of a 2-dimensional disc in the Euclidean plane is +∞; likewise, μ s (E) = 0 for s > dim H (E) for the same reason as the 3-dimensional 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) s for some s, and to look for the same property of the Hausdorff measure being finite and non-zero. ==Definition== Let (X, d) be a metric space and E ⊆ X. Let h : [0, +∞) → [0, +∞] be a function. Define μ h (E) by \\mu^{h} (E) = \\lim_{\\delta \\to 0} \\mu_{\\delta}^{h} (E), where \\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\\}. Then h is called an (exact) dimension function (or gauge function) for E if μ h (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 for t ≥ 0, strictly positive for t > 0, and continuous on the right for all t ≥ 0. ===Packing dimension=== Packing dimension is constructed in a very similar way to Hausdorff dimension, except that one \"packs\" E from inside with pairwise disjoint balls of diameter at most δ. Just as before, one can consider functions h : [0, +∞) → [0, +∞] more general than h(δ) = δ s and call h an exact dimension function for E if the h-packing measure of E is finite and strictly positive. ==Example== Almost surely, a sample path X of Brownian motion in the Euclidean plane has Hausdorff dimension equal to 2, but the 2-dimensional Hausdorff measure μ 2 (X) is zero. The exact dimension function h is given by the logarithmic correction h(r) = r^{2} \\cdot \\log \\frac1{r} \\cdot \\log \\log \\log \\frac1{r}. I.e., with probability one, 0 < μ h (X) < +∞ for a Brownian path X in R 2 . For Brownian motion in Euclidean n-space R n with n ≥ 3, the exact dimension function is h(r) = r^{2} \\cdot \\log \\log \\frac1r. ==References==",
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