On levels in arrangements of lines, segments, planes, and triangles

We consider the problem of bounding the complexity of the kth level in an arrangement of n curves or surfaces, a problem dual to, and an extension of, the well-known k-set problem. Among other results, we prove a new bound, O(nk^5/3), on the complexity of the kth level in an arrangement of n planes in ℝ³, or on the number of k-sets in a set of n points in three dimensions, and we show that the complexity of the kth level in an arrangement of n line segments in the plane is O(n-√kα(n/k)), and that the complexity of the kth level in an arrangement of n triangles in 3-space is O(n²k^5/6α(n/k)).