On the union of κ-round objects in three and four dimensions

Boris Aronov, Alon Efrat, Vladlen Koltun, Micha Sharir

Research output: Contribution to journalArticlepeer-review

10 Scopus citations


A compact set c in ℝd is κ-round if for every point p ∈ ∂ C there exists a closed ball that contains p, is contained in c, and has radius κ diam c. We show that, for any fixed κ > 0, the combinatorial complexity of the union of n κ-round, not necessarily convex, objects in ℝ3 (resp., in ℝ4) of constant description complexity is O(n2+ε) (resp., O(n 3+ε)) for any ε > 0, where the constant of proportionality depends on ε, κ, and the algebraic complexity of the objects. The bound is almost tight in the worst case.

Original languageEnglish (US)
Pages (from-to)511-526
Number of pages16
JournalDiscrete and Computational Geometry
Issue number4
StatePublished - Dec 2006

ASJC Scopus subject areas

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


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