Discrete & Computational Geometry

, Volume 12, Issue 3, pp 327–345 | Cite as

Almost tight upper bounds for lower envelopes in higher dimensions

  • M. Sharir


We consider the problem of bounding the combinatorial complexity of the lower envelope ofn surfaces or surface patches ind-space (d≥3), all algebraic of constant degree, and bounded by algebraic surfaces of constant degree. We show that the complexity of the lower envelope ofn such surface patches isO(nd−1+∈), for any ∈>0; the constant of proportionality depends on ∈, ond, ons, the maximum number of intersections among anyd-tuple of the given surfaces, and on 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 isO(nd-2λ q (n)) for some constantq 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 timeO(n2+∈), and give several applications of the new bounds.


Voronoi Diagram Surface Patch Algebraic Surface Intersection Curve Relative Interior 
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Copyright information

© Springer-Verlag New York Inc. 1994

Authors and Affiliations

  • M. Sharir
    • 1
    • 2
  1. 1.School of Mathematical SciencesTel Aviv UniversityTel AvivIsrael
  2. 2.Courant Institute of Mathematical Sciences, New York UniversityNew YorkUSA

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