Minkowski content

The Minkowski content (named after Hermann Minkowski), or the boundary measure, of a set is a basic concept that uses concepts from geometry and measure theory to generalize the notions of length of a smooth curve in the plane, and area of a smooth surface in the space, to arbitrary measurable sets.

It is typically applied to fractal boundaries of domains in the Euclidean space, but it can also be used in the context of general metric measure spaces.

It is related to, although different from, the Hausdorff measure.

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

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