Definition:Hausdorff-Besicovitch Dimension
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Definition
Let $\R^n$ be the $n$-dimensional Euclidean space.
Let $F \subseteq \R^n$.
The Hausdorff-Besicovitch dimension of $F$ is defined as:
\(\ds \map {\dim_H} F\) | \(:=\) | \(\ds \inf \set {s \in \R_{\ge 0} : \map {\HH^s} F = 0}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sup \set {s \in \R_{\ge 0} : \map {\HH^s} F = +\infty}\) |
where $\map {\HH^s} \cdot$ denotes the $s$-dimensional Hausdorff measure on $\R^n$.
Also known as
The Hausdorff-Besicovitch dimension is also (and usually) known as the Hausdorff dimension.
Also see
- Hausdorff-Besicovitch Dimension is Well-Defined : In particular, it is shown that $\map {\dim_H} F \in \closedint 0 n$.
- Definition:Box-Counting Dimension
- Definition:Packing Dimension
- Results about the Hausdorff-Besicovitch dimension can be found here.
Source of Name
This entry was named for Felix Hausdorff and Abram Samoilovitch Besicovitch.
Historical Note
The Hausdorff-Besicovitch dimension was introduced by Felix Hausdorff in $1919$.
Sources
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Hausdorff dimension
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): fractal
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Hausdorff dimension
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): fractal
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Hausdorff dimension
- 2014: Kenneth Falconer: Fractal Geometry: Mathematical Foundations and Applications (3rd ed.): $3.1$ Hausdorff measure