Discrete & Computational Geometry

, Volume 4, Issue 2, pp 139–181

The complexity of cutting complexes

  • Bernard Chazelle
  • Herbert Edelsbrunner
  • Leonidas J. Guibas

DOI: 10.1007/BF02187720

Cite this article as:
Chazelle, B., Edelsbrunner, H. & Guibas, L.J. Discrete Comput Geom (1989) 4: 139. doi:10.1007/BF02187720


This paper investigates the combinatorial and computational aspects of certain extremal geometric problems in two and three dimensions. Specifically, we examine the problem of intersecting a convex subdivision with a line in order to maximize the number of intersections. A similar problem is to maximize the number of intersected facets in a cross-section of a three-dimensional convex polytope. Related problems concern maximum chains in certain families of posets defined over the regions of a convex subdivision. In most cases we are able to prove sharp bounds on the asymptotic behavior of the corresponding extremal functions. We also describe polynomial algorithms for all the problems discussed.

Copyright information

© Springer-Verlag New York Inc. 1989

Authors and Affiliations

  • Bernard Chazelle
    • 1
  • Herbert Edelsbrunner
    • 2
  • Leonidas J. Guibas
    • 3
  1. 1.Department of Computer SciencePrinceton UniversityPrincetonUSA
  2. 2.Department of Computer ScienceUniversity of IllinoisUrbanaUSA
  3. 3.Department of Computer ScienceStanford University/DEC-SRCPalo AltoUSA