On computing the exact value of dispersion of a sequence (Q1200177)

This paper introduces a method for computing the exact value of the dispersion, which is a well-known measure for the irregularity of distribution of sequences and defined for the sequence \(G_ N=\{x_ 1,\dots,x_ N\}\) in \(I^ N\) by \(d(G_ N)=\max_{1\leq i\leq N} \max_{x\in S(i)} d(x,x_ i)\) where \(S(i)=\{x\in I^ s\): \(d(x,x_ i)\leq d(x,x_ k)\), \(k=1,2,\dots,N\}\). Recognizing that the problem of finding the extreme points of \(S(i)\) can be simplified by eliminating the redundant inequalities from the set of inequalities which define \(S(i)\), the authors obtain the results useful to identify the redundant constraints in \(\mathbb{R}^ 2\). Then they obtain an explicit formula for the dispersion of the two-dimensional Hammersley sequence of \(\mathbb{R}^ N\) points, where \(N\) is a positive integer.