Definition:Hausdorff-Besicovitch Dimension

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