Minkowski content

The Minkowski content of a set (named after Hermann Minkowski), or the boundary measure, is a basic concept in geometry and measure theory which generalizes to arbitrary measurable sets the notions of length of a smooth curve in the plane and area of a smooth surface in the space. It is typically applied to fractal boundaries of domains in the Euclidean space, but makes sense in the context of general metric measure spaces. It is related to, although different from, the Hausdorff measure.

Definition

Let $\scriptstyle(X,\, \mu,\, d)$ be a metric measure space, where d is a metric on X and μ is a Borel measure. For a subset A of X and real ε > 0, let

$A_\varepsilon = \{ x \in X \, | \, d(x, A) \leq \varepsilon \}$

be the ε-extension of A. The lower Minkowski content of A is given by

$M_*(A) = \liminf_{\varepsilon \to 0} \frac{\mu(A_\varepsilon) - \mu(A)}{\varepsilon},$

and the upper Minkowski content of A is

$M^*(A) = \limsup_{\varepsilon \to 0} \frac{\mu(A_\varepsilon) - \mu(A)}{\varepsilon}.$

If M_*(A) = M^*(A), then the common value is called the Minkowski content of A associated with the measure μ, and is denoted by M(A).

Minkowski content in Rⁿ

Let A be a subset of Rⁿ. Then the m-dimensional Minkowski content of A is defined as follows. The lower content is

$M_*^m(A) = \liminf_{\varepsilon \to 0} \frac{\mu(A_\varepsilon) - \mu(A)}{\alpha_{n-m}\varepsilon^{n-m}}$

where α_n−m is the volume of the unit (n−m)-ball and $\mu$ is $n$-dimensional Lebesgue measure. The upper content is

$M^{*m}(A) = \limsup_{\varepsilon \to 0} \frac{\mu(A_\varepsilon) - \mu(A)}{\alpha_{n-m}\varepsilon^{n-m}}.$

As before, if the upper and lower m-dimensional Minkowski content of A agree, then the Minkowski content of A, M^m(A), is defined to be this common value.

Properties

• The Minkowski content is (generally) not a measure. In particular, the m-dimensional Minkowski content in Rⁿ is not a measure unless m = 0, in which case it is the counting measure. Indeed, clearly the Minkowski content assigns the same value to the set A as well as its closure.
• If A is a closed m-rectifiable set in Rⁿ, given as the image of a bounded set from R^m under a Lipschitz function, then the m-dimensional Minkowski content of A exists, and is equal to the m-dimensional Hausdorff measure of A, apart from a constant normalization depending on the dimension.

See also

• Gaussian isoperimetric inequality
• Geometric measure theory
• Isoperimetric problem
• Minkowski–Bouligand dimension

References

• Federer, Herbert (1969), Geometric Measure Theory, Springer-Verlag, ISBN 3-540-60656-4.
• Krantz, Steven G.; Parks, Harold R. (1999), The geometry of domains in space, Birkhäuser Advanced Texts: Basler Lehrbücher, Boston, MA: Birkhäuser Boston, Inc., ISBN 0-8176-4097-5, MR 1730695.