Many-Face Complexity in Incremental Convex Arrangements

Abstract We consider the many-face problem in incremental convex arrangements of plates on hyperplanes. An incremental convex arrangement of plates is built up by inserting plates incrementally producing only convex cells. We show that the complexity of m interior-disjoint convex d-polytopes in such arrangements is Θ (mn) if m ⩽ n and O(m(1−1/d)·n(d −2)(1+1/d)) if m ⩾ n. For certain values of m, one can derive an improved bound using ϵ-cuttings.