On the overlay of envelopes in four dimensions

We show that the complexity of the overlay of two envelopes of arrangements of n semi-algebraic surfaces or surface patches of constant description complexity in four dimensions is 0(n4-1/|s/2+ϵ), for any ϵ > 0, where s is a constant related to the maximal degree of the surfaces. This is the first non-trivial (sub-quartic) bound for this problem, and for s = 1,2 it almost matches the near-cubic lower bound. We discuss several applications of this result, including (i) an improved bound for the complexity of the region enclosed between two envelopes in four dimensions, (ii) an improved bound for the complexity of the space of all hyperplane transversals of a collection of simply-shaped convex sets in 4-space, (iii) an improved bound for the complexity of the space of all line transversals of a similar collection of sets in 3-space, and (iv) improved bounds for the complexity of the union of certain families of objects in four dimensions. The analysis technique we introduce is quite general, and has already proved useful in unrelated contexts.