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 space, to arbitrary measurable sets.
It is related to, although different from, the Hausdorff measure.
For , and each integer m with , the m-dimensional upper Minkowski content is
and the m-dimensional lower Minkowski content is defined as
- 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 Rm 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.
- Gaussian isoperimetric inequality
- Geometric measure theory
- Isoperimetric inequality in higher dimensions
- Minkowski–Bouligand dimension