Minkowski content

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The Minkowski content of a set, 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.

[edit] 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 boundary measure of A in the sense of Minkowski associated with μ is the nonnegative real number \scriptstyle \mu^+(A) defined as follows:

\mu^+(A) = \liminf_{\varepsilon \to 0} \frac{\mu(A_\varepsilon) - \mu(A)}{2\varepsilon}.

[edit] See also

[edit] References

  • Steven G. Krantz and Harold R. Parks, The geometry of domains in space. Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Boston, Inc., Boston, MA, 1999 ISBN 0-8176-4097-5 MR1730695