Geometry of Euclidean Sets: Analytical, Stochastic and Hypertopological Aspects : Theory of Dimension II: “Metric” Dimensions

Theory of Dimension II: “Metric” Dimensions
Geometry of Euclidean Sets: Analytical, Stochastic and Hypertopological Aspects

Author : Jean-Charles PINOLI

Publication date: June 10, 2026 | Lire en français

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2. Theory of Dimension II: “Metric” Dimensions

Dimension Theory is an interdisciplinary branch of mathematics related in particular to Geometric Measure Theory.

2.1 Hausdorff-Besicovitch Dimension

The Hausdorff-Besicovitch dimension is useful for studying structurally complex sets, particularly fractals (see below). Unlike the dimensions discussed above, the Hausdorff dimension can also take on non-integer real values.

Definition (Hausdorff-Besicovitch dimension). The Hausdorff-Besicovitch dimension of a nonempty subset X of $ℝ_{2}^{n}$ is a positive real number, zero, or possibly infinity, denoted by dim_HB(X), defined by:

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Theory of Dimension II: “Metric” Dimensions