Minkowski content

For ${\displaystyle A\subset \mathbb {R} ^{n}}$, and each integer m with ${\displaystyle 0\leq m\leq n}$, the m-dimensional upper Minkowski content is

${\displaystyle M^{*m}(A)=\limsup _{r\to 0^{+}}{\frac {\mu (\{x:d(x,A)<r\})}{\alpha (n-m)r^{n-m}}}}$

and the m-dimensional lower Minkowski content is defined as

${\displaystyle M_{*}^{m}(A)=\liminf _{r\to 0^{+}}{\frac {\mu (\{x:d(x,A)<r\})}{\alpha (n-m)r^{n-m}}}}$

where ${\displaystyle \alpha (n-m)r^{n-m}}$ is the volume of the (n−m)-ball of radius r and ${\displaystyle \mu }$ is an ${\displaystyle n}$-dimensional Lebesgue measure.

If the upper and lower m-dimensional Minkowski content of A are equal, then their common value is called the Minkowski content M^m(A).[1][2]

• 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.[3]

• Gaussian isoperimetric inequality
• Geometric measure theory
• Isoperimetric inequality in higher dimensions
• Minkowski–Bouligand dimension

• 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.