An L~1 estimate for half-space discrepancy by WILLIAM W. L. CHEN (Sydney) and GIANCARLO TRAVAGLINI (Milano)

摘要

The half-space discrepancy is a typical problem in the study of irregularities of point distribution, and represents a multidimensional variant of an open problem first posed by Roth; see Schmidt [8, pp. 124-125]. In its general form, it asks whether it is possible to choose N points in a given bounded convex body in such a way that after cutting it into two parts by hyperplanes in different ways, the numbers of points in the two parts essentially depend only on the relative volumes. More precisely, let P denote a distribution of N points in a bounded convex body B ∈ R~d. For every unit vector σ∈Σ _(d-1) and every r ≥ 0, consider the half-space H_(σ,r)={t∈R~d:t·σ≤ r}, where· denotes the usual inner product in R~d, and let S_(σ, r) = B∩H_(σ,r). The problem is whether (1)(1.1) inf sup |card(P∩S_(σ,r)) — N|B|~(-1)|S_(σ,r|| card(P)=N r>0σ∈Σ_(d-1)is unbounded with N.