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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 ...
The Minkowski distance can also be viewed as a multiple of the power mean of the component-wise differences between and . The following figure shows unit circles (the level set of the distance function where all points are at the unit distance from the center) with various values of p {\displaystyle p} :
The reverse inequality follows from the same argument as the standard Minkowski, but uses that Holder's inequality is also reversed in this range. Using the Reverse Minkowski, we may prove that power means with , such as the harmonic mean and the geometric mean are concave.
A Minkowski diagram is a two-dimensional graphical depiction of a portion of Minkowski space, usually where space has been curtailed to a single dimension. The units of measurement in these diagrams are taken such that the light cone at an event consists of the lines of slope plus or minus one through that event. [ 3 ]
The second part begins with Minkowski's theorem, that centrally symmetric convex sets of large enough area (or volume in higher dimensions) necessarily contain a nonzero lattice point. It applies this to Diophantine approximation , the problem of accurately approximating one or more irrational numbers by rational numbers.
Myrheim–Meyer dimension This approach relies on estimating the number of k {\displaystyle k} -length chains present in a sprinkling into d {\displaystyle d} -dimensional Minkowski spacetime. Counting the number of k {\displaystyle k} -length chains in the causal set then allows an estimate for d {\displaystyle d} to be made.
A probability measure is a finite measure, and the Shapley–Folkman lemma has applications in non-probabilistic measure theory, such as the theories of volume and of vector measures. The Shapley–Folkman lemma enables a refinement of the Brunn–Minkowski inequality , which bounds the volume of sums in terms of the volumes of their summand ...