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 space, to arbitrary measurable sets. 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 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 A ⊂ R n {displaystyle Asubset mathbb {R} ^{n}} , and each integer m with 0 ≤ m ≤ n {displaystyle 0leq mleq n} , the m-dimensional upper Minkowski content is and the m-dimensional lower Minkowski content is defined as where α ( n − m ) r n − m {displaystyle alpha (n-m)r^{n-m}} is the volume of the (n−m)-ball of radius r and μ {displaystyle mu } is an n {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 Mm(A).