On the complexity of the k-level in arrangements of pseudoplanes

A classical open problem in combinatorial geometry is to obtain tight asymptotic bounds on the maximum number of k-level vertices in an arrangement of n hyperplanes in R^d (vertices with exactly k of the hyperplanes passing below them). This is essentially a dual version of the k-set problem, which, in a primal setting, seeks bounds for the maximum number of k-sets determined by n points in R^d, where a k-set is a subset of size k that can be separated from its complement by a hyperplane. The k-set problem is still wide open even in the plane. In three dimensions, the best known upper and lower bounds are, respectively, O(nk³/²) [15] and nk · 2^Ω(log k⁾ [19]. In its dual version, the problem can be generalized by replacing hyperplanes by other families of surfaces (or curves in the planes). Reasonably sharp bounds have been obtained for curves in the plane [16, 18], but the known upper bounds are rather weak for more general surfaces, already in three dimensions, except for the case of triangles [1]. The best known general bound, due to Chan [7] is O(n^2.997), for families of surfaces that satisfy certain (fairly weak) properties. In this paper we consider the case of pseudoplanes in R³ (defined in detail in the introduction), and establish the upper bound O(nk⁵/³) for the number of k-level vertices in an arrangement of n pseudoplanes. The bound is obtained by establishing suitable (and nontrivial) extensions of dual versions of classical tools that have been used in studying the primal k-set problem, such as the Lovász Lemma and the Crossing Lemma.