Minkowski content
From Wikipedia, the free encyclopedia
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 
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
be the ε-extension of A. The boundary measure of A in the sense of Minkowski associated with μ is the nonnegative real number 
defined as follows:
[edit] See also
- Gaussian isoperimetric inequality
- Geometric measure theory
- Isoperimetric problem
- Minkowski-Bouligand dimension
- Rectifiable
[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
