Computing Hereditary Convex Structures

Bernard Chazelle, Wolfgang Mulzer

Research output: Contribution to journalArticlepeer-review

7 Scopus citations


Color red and blue the n vertices of a convex polytope P in ℝ3. Can we compute the convex hull of each color class in o(n log n) time? What if we have more than two colors? What if the colors are random? Consider an arbitrary query halfspace and call the vertices of P inside it blue: can the convex hull of the blue points be computed in time linear in their number? More generally, can we quickly compute the blue hull without looking at the whole polytope? This paper considers several instances of hereditary computation and provides new results for them. In particular, we resolve an eight-year old open problem by showing how to split a convex polytope in linear expected time.

Original languageAmerican English
Pages (from-to)796-823
Number of pages28
JournalDiscrete and Computational Geometry
Issue number4
StatePublished - Jun 2011

ASJC Scopus subject areas

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


  • Convex polytope
  • Halfspace range searching
  • Hereditary convex hulls

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