The complexity of many cells in arrangements of planes and related problems

We consider several problems involving points and planes in three dimensions. Our main results are: (i) The maximum number of faces bounding m distinct cells in an arrangement of n planes is O(m^2/3n log n +n²); we can calculate m such cells specified by a point in each, in worst-case time O(m^2/3n log³n+n² log n). (ii) The maximum number of incidences between n planes and m vertices of their arrangement is O(m^2/3n log n+n²), but this number is only O(m^3/5-δn^{4/5+2 δ}+m+n log m), for any δ>0, for any collection of points no three of which are collinear. (iii) For an arbitrary collection of m points, we can calculate the number of incidences between them and n planes by a randomized algorithm whose expected time complexity is O((m^3/4-δn^{3/4+3 δ}+m) log²n+n log n log m) for any δ>0. (iv) Given m points and n planes, we can find the plane lying immediately below each point in randomized expected time O([m^3/4-δn^{3/4+3 δ}+m] log²n+n log n log m) for any δ>0. (v) The maximum number of facets (i.e., (d-1)-dimensional faces) bounding m distinct cells in an arrangement of n hyperplanes in d dimensions, d>3, is O(m^2/3n^d/3 log n+n^d-1). This is also an upper bound for the number of incidences between n hyperplanes in d dimensions and m vertices of their arrangement. The combinatorial bounds in (i) and (v) and the general bound in (ii) are almost tight.