Polytopes in arrangements

Consider an arrangement of n hyperplanes in ℝ^d. Families of convex polytopes whose boundaries are contained in the union of the hyperplanes are the subject of this paper. We aim to bound their maximum combinatorial complexity. Exact asymptotic bounds were known for the case where the polytopes are cells of the arrangement. Situations where the polytopes are pairwise openly disjoint have also been considered in the past. However, no nontrivial bound was known for the general case where the polytopes may have overlapping interiors, for d > 2. We analyze families of polytopes that do not share vertices. In ℝ³ we show an O (k^1/3n²) bound on the number of faces of k such polytopes. We also discuss worst-case lower bounds and higher-dimensional versions of the problem. Among other results, we show that the maximum number of facets of k pairwise vertex-disjoint polytopes in ℝ^d is Ω (k^1/2n^d/2) which is a factor of √n away from the best known upper bound in the range n^d-2 ≤ k ≤ n^d. The case where 1 ≤ k ≤ n^d-2 is completely resolved as a known Θ(kn) bound for cells applies here.