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Estimating the box-counting dimension of the coast of Great Britain. In fractal geometry, the Minkowski–Bouligand dimension, also known as Minkowski dimension or box-counting dimension, is a way of determining the fractal dimension of a bounded set in a Euclidean space, or more generally in a metric space (,).
In particular, the m-dimensional Minkowski content in R n 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 R n, given as the image of a bounded set from R m under a Lipschitz function ...
For any -dimensional polytope, one can specify its collection of facet directions and measures by a finite set of -dimensional nonzero vectors, one per facet, pointing perpendicularly outward from the facet, with length equal to the ()-dimensional measure of its facet. [3]
The Minkowski dimension is similar to, and at least as large as, the Hausdorff dimension, and they are equal in many situations. However, the set of rational points in [0, 1] has Hausdorff dimension zero and Minkowski dimension one. There are also compact sets for which the Minkowski dimension is strictly larger than the Hausdorff dimension.
In mathematical analysis, the Minkowski inequality establishes that the L p spaces are normed vector spaces. Let be a measure space, let < and let ...
Let = [,] denote the unit interval. Note that the box-counting dimension and the Minkowski dimension coincide with a common value of 1; i.e. = = Now observe that (,) = ⌊ / ⌋ +, where ⌊ ⌋ denotes the integer part of .
In full generality, the Minkowski problem asks for necessary and sufficient conditions on a non-negative Borel measure on the unit sphere S n-1 to be the surface area measure of a convex body in . Here the surface area measure S K of a convex body K is the pushforward of the (n-1) -dimensional Hausdorff measure restricted to the boundary of K ...
When both polytopes have parallel facets, the measure of the corresponding facet in the Blaschke sum is the sum of the measures from the two given polytopes. [ 1 ] Blaschke sums exist and are unique up to translation , as can be proven using the theory of the Minkowski problem for polytopes .