Discrete & Computational Geometry

, Volume 4, Issue 2, pp 139-181

First online:

The complexity of cutting complexes

  • Bernard ChazelleAffiliated withDepartment of Computer Science, Princeton University
  • , Herbert EdelsbrunnerAffiliated withDepartment of Computer Science, University of Illinois
  • , Leonidas J. GuibasAffiliated withDepartment of Computer Science, Stanford University/DEC-SRC

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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.