The complexity of cutting complexes

Bernard Chazelle, Herbert Edelsbrunner, Leonidas J. Guibas

Research output: Contribution to journalArticlepeer-review

53 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)139-181
Number of pages43
JournalDiscrete & Computational Geometry
Issue number1
StatePublished - Dec 1989

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics


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