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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 set]]s the notions of [[arc length|length]] of a [[smooth curve]] in the plane and [[surface area|area]] of a smooth surface in the [[Euclidean space|space]]. It is typically applied to [[fractal]] boundaries of domains in the [[Euclidean space]], but makes sense in the context of general metric [[Measure space|measure spaces.]]  It is related to, although different from, the [[Hausdorff measure]].


== Definition ==
Let <math>\scriptstyle(X,\, \mu,\, d)</math> be a metric measure space, where ''d'' is a [[metric (mathematics)|metric]] on ''X'' and &mu; is a [[Borel measure]]. For a subset ''A'' of ''X'' and real &epsilon; > 0, let


: <math>A_\varepsilon = \{ x \in X \, | \, d(x, A) \leq \varepsilon \}</math>
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be the &epsilon;-''extension'' of ''A''. The '''lower Minkowski content''' of ''A'' is given by
 
:<math>M_*(A) = \liminf_{\varepsilon \to 0} \frac{\mu(A_\varepsilon) - \mu(A)}{2\varepsilon},</math>
 
and the '''upper Minkowski content''' of ''A'' is
 
:<math>M^*(A) = \limsup_{\varepsilon \to 0} \frac{\mu(A_\varepsilon) - \mu(A)}{2\varepsilon}.</math>
 
If ''M''<sub>*</sub>(''A'')&nbsp;=&nbsp;''M''<sup>*</sup>(''A''), then the common value is called the '''Minkowski content''' of ''A'' associated with the measure &mu;, and is denoted by ''M''(''A'').
 
== Minkowski content in '''R'''<sup>''n''</sup> ==
Let ''A'' be a subset of '''R'''<sup>''n''</sup>.  Then the '''''m''-dimensional Minkowski content''' of ''A'' is defined as follows. The lower content is
 
:<math>M_*^m(A) = \liminf_{\varepsilon \to 0} \frac{\mu(A_\varepsilon) - \mu(A)}{\alpha_{n-m}\varepsilon^{n-m}}</math>
 
where &alpha;<sub>''n''&minus;''m''</sub> is the volume of the unit [[N-sphere|(''n''&minus;''m'')-ball]] and <math>\mu</math> is <math>n</math>-dimensional [[Lebesgue measure]].  The upper content is
 
:<math>M^{*m}(A) = \limsup_{\varepsilon \to 0} \frac{\mu(A_\varepsilon) - \mu(A)}{\alpha_{n-m}\varepsilon^{n-m}}.</math>
 
As before, if the upper and lower ''m''-dimensional Minkowski content of ''A'' agree, then the Minkowski content of ''A'', ''M''<sup>''m''</sup>(''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 '''R'''<sup>n</sup> is not a measure unless ''m''&nbsp;=&nbsp;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 (topology)|closure]].
* If ''A'' is a closed ''m''-[[rectifiable set]] in '''R'''<sup>''n''</sup>, given as the image of a bounded set from '''R'''<sup>''m''</sup> 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 ==
* {{citation|first=Herbert|last=Federer|authorlink=Herbert Federer|title=Geometric Measure Theory|publisher=Springer-Verlag|year=1969|isbn=3-540-60656-4}}.
* {{citation|first1=Steven G.|last1=Krantz|first2=Harold R.|last2=Parks|author2-link=Harold R. Parks|title=The geometry of domains in space|series=Birkhäuser Advanced Texts: Basler Lehrbücher|publisher=Birkhäuser Boston, Inc.|publication-place=Boston, MA|year=1999|isbn=0-8176-4097-5|id={{MathSciNet |id=1730695}} }}.
 
[[Category:Measures (measure theory)]]
[[Category:Fractals]]

Latest revision as of 00:41, 20 October 2014


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