Almost tight upper bounds for lower envelopes in higher dimensions

We show that the combinatorial complexity of the lower envelope of n surfaces or surface patches in d-space (d⩾3), all algebraic of constant maximum degree, and bounded by algebraic surfaces of constant maximum degree, is O(n^d-1+ε), for any ε>0; the constant of proportionality depends on ε, d, and the shape and degree of the surface patches and of their boundaries. This is the first nontrivial general upper bound for this problem, and it almost establishes a long-standing conjecture that the complexity of the envelope is O(n^d-2λ_q(n)) for some constant q depending on the shape and degree of the surfaces (where λ_q(n) is the maximum length of (n,q) Davenport-Schinzel sequences). We also present a randomized algorithm for computing the envelope in three dimensions, with expected running time O(n^2+ε), and give several applications of the new bounds