Suppose that equal numbers of
red and blue points, all distinct, lie in the euclidean space Et, and consider a
hyperplane h containing none of the points. If H is one of the open halfspaces
determined by h, let D(h) denote |r(h) − b(h)| where r(h) and b(h) are the
numbers of red and blue points lying in H. What can be said about the number
supD(h) as h ranges over all hyperplanes? The present article addresses this
and similar problems of discrepancy principally by developing estimates of
L2 integral averages of D(h) with respect to the invariant measure on the
planesets of Et. Special attention is given to the influence of the dimension
t.
The aim is to develop inequalities that involve only absolute constants and simple
geometric properties of a given pointmass distribution. For example, the
following theorem is an immediate corollary to more general results in this
article.
