The exact fitting problem in higher dimensions

Let S be a family of n points in E(d). The exact fitting problem is that of finding a hyperplane containing the maximum number of points of S. In this paper, we present an O(min{(n(d)/m(d-1)) log(n/m), n(d)}) time algorithm where m denotes the number of points in the hyperplane. This algorithm is based on upper bounds on the maximum number of incidences between families of points and families of hyperplanes in E(d) and on an algorithm to compute these incidences. We also show how the upper bound on the maximum number of incidences between families of points and families of hyperplanes can be used to derive new bounds on some well-known problems in discrete geometry.