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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]
Minkowski weighted k-means automatically calculates cluster specific feature weights, supporting the intuitive idea that a feature may have different degrees of relevance at different features. [42] These weights can also be used to re-scale a given data set, increasing the likelihood of a cluster validity index to be optimized at the expected ...
In mathematical analysis, the Minkowski inequality establishes that the L p spaces are normed vector spaces. Let be a measure space, let < and let ...
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 ...
If is a subset of a real or complex vector space, then the Minkowski functional or gauge of is defined to be the function: [,], valued in the extended real numbers, defined by ():= {: >}, where the infimum of the empty set is defined to be positive infinity (which is not a real number so that () would then not be real-valued).
Hermann Minkowski (1864–1909) found that the theory of special relativity could be best understood as a four-dimensional space, since known as the Minkowski spacetime.. In physics, Minkowski space (or Minkowski spacetime) (/ m ɪ ŋ ˈ k ɔː f s k i,-ˈ k ɒ f-/ [1]) is the main mathematical description of spacetime in the absence of gravitation.