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 dspace ( d 2 3), all algebraic of constant maximum degree, and bounded by algebraic surfaces of constant maximum degree, is O(rkl+') , for any E > 0; the constant of proportionality depends on E , 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(nd-2X,(n) ) for some constant. q depending on the shape and degree of the surfaces (where X,(n) is the maximum length of ( n , q ) DavenportSchinzel sequences). We also present a randomized algorithm for computing the envelope in three dimensions, with expect,ed running time O(n?+'), a.nd give several applications of the new bounds.