Definition:Hausdorff-Besicovitch Dimension

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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

This is also known as the Hausdorff dimension.

Also see

Source of Name

This entry was named for Felix Hausdorff and Abram Samoilovitch Besicovitch.