On levels in arrangements of surfaces in three dimensions

A favorite open problem in combinatorial geometry is to determine the worst-case complexity of a level in an arrangement. Up to now, nontrivial upper bounds in three dimensions are known only for the linear cases of planes and triangles. We propose the first technique that can deal with more general surfaces in three dimensions. For example, in an arrangement of n "pseudo-planes" or "pseudo-spherical patches" (where the main criterion is that each triple of surfaces has at most two common intersections), we prove that there are at most O(n ^2.997) vertices at any given level.