Irregularities of point distributions relative to homothetic convex bodies I

The uniformity and irregularities of point distributions can be measured by various kinds of geometric objects. In this paper we prove the existence of point sets that have relatively small irregularities with respect to homothetic copies of a fixed convex body. The results give higher dimensional alternatives to a theorem of Beck.

[1]  K. F. Roth.  On irregularities of distribution , 1954 .

[2]  J. Halton.  On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals , 1960 .

[3]  E. Hlawka.  Funktionen von beschränkter Variatiou in der Theorie der Gleichverteilung , 1961 .

[4]  Wolfgang M. Schmidt.  Irregularities of distribution. IV , 1969 .

[5]  P. McMullen.  The maximum numbers of faces of a convex polytope , 1970 .

[6]  G. C. Shephard,et al.  Convex Polytopes and the Upper Bound Conjecture , 1971 .

[7]  W. Schmidt.  On irregularities of distribution vii , 1972 .

[8]  Lauwerens Kuipers,et al.  Uniform distribution of sequences , 1974 .

[9]  W. Schmidt.  Irregularities for distribution IX , 1975 .

[10]  W. Stute.  Convergence Rates for the Isotrope Discrepancy , 1977 .

[11]  H. Niederreiter.  Quasi-Monte Carlo methods and pseudo-random numbers , 1978 .

[12]  L. Lovász,et al.  Geometric Algorithms and Combinatorial Optimization , 1981 .

[13]  Louis J. Billera,et al.  Counting Faces and Chains in Polytopes and Posets , 1983 .

[14]  J. Beck,et al.  Irregularities of distribution , 1987 .

[15]  H. Niederreiter.  Low-discrepancy and low-dispersion sequences , 1988 .

[16]  On the discrepancy of convex plane sets , 1988 .

[17]  J. Beck.  Irregularities of Distribution, Ii , 1988 .

[18]  J. Beck.  A two-dimensional van Aardenne-Ehrenfest theorem in irregularities of distribution , 1989 .

[19]  N. Biggs.  GEOMETRIC ALGORITHMS AND COMBINATORIAL OPTIMIZATION: (Algorithms and Combinatorics 2) , 1990 .

[20]  József Beck,et al.  Irregularities of point distribution relative to convex polygons II , 1993 .

[21]  G. Károlyi.  Geometric discrepancy theorems in higher dimensions , 1995 .