Irregularities of point distributions relative to homothetic convex bodies. I
Let \(A\subset \mathbb{R}^k\) be a convex body and \(\mu\) the \(k\)-dimensional Lebesgue measure. For an arbitrary distribution \(P= \{p_1, \dots, p_N\}\) of \(N\) points in the unit cube \([0,1)^k\) let \(\Delta (P, A)= \sup|Z(P, (\lambda A+v)\cap [0,1 )^k- N\mu (\lambda A)|\), where the supremum is extended over all real numbers \(-1\leq \lambda\leq 1\) and all vectors \(v\in \mathbb{R}^k\). In the two-dimensional case \textit{J. Beck} [Proc. Lond. Math. Soc., III. Ser. 56, 1-50 (1988; Zbl 0644.10038)]\ has proved an upper bound for \(\Delta (P,A)\). In the present paper this upper bound is extended to higher dimensions. The main tool is a geometric lemma which transforms the problem to a 2-coloring problem of a hypergraph.
scientific article; zbMATH DE number 838722