Search results
Results From The WOW.Com Content Network
An interesting property of the upper box dimension not shared with either the lower box dimension or the Hausdorff dimension is the connection to set addition. If A and B are two sets in a Euclidean space, then A + B is formed by taking all the pairs of points a , b where a is from A and b is from B and adding a + b .
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 distance or Minkowski metric is a metric in a normed vector space which can be considered as a generalization of both the Euclidean distance and the ...
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 ...
A set in ℝ 2 satisfying the hypotheses of Minkowski's theorem. In mathematics , Minkowski's theorem is the statement that every convex set in R n {\displaystyle \mathbb {R} ^{n}} which is symmetric with respect to the origin and which has volume greater than 2 n {\displaystyle 2^{n}} contains a non-zero integer point (meaning a point in Z n ...
In geometry, the hyperboloid model, also known as the Minkowski model after Hermann Minkowski, is a model of n-dimensional hyperbolic geometry in which points are represented by points on the forward sheet S + of a two-sheeted hyperboloid in (n+1)-dimensional Minkowski space or by the displacement vectors from the origin to those points, and m ...