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/spl ges/3), all algebraic of constant maximum degree, and bounded by algebraic surfaces of constant maximum degree, is O(n/sup d-1+/spl epsi//), for any /spl epsi/>0; the constant of proportionality depends on /spl epsi/, 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/sup d-2//spl lambda//sub q/(n)) for some constant q depending on the shape and degree of the surfaces (where /spl lambda//sub 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/sup 2+/spl epsi//), and give several applications of the new bounds.<
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