### Abstract

A compact body c in ℝ^{d} 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 ℝ^{3} (resp., in ℝ^{4}) of constant description complexity is O(n^{2+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.

Original language | English (US) |
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Pages | 383-390 |

Number of pages | 8 |

State | Published - Sep 29 2004 |

Event | Proceedings of the Twentieth Annual Symposium on Computational Geometry (SCG'04) - Brooklyn, NY, United States Duration: Jun 9 2004 → Jun 11 2004 |

### Other

Other | Proceedings of the Twentieth Annual Symposium on Computational Geometry (SCG'04) |
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Country | United States |

City | Brooklyn, NY |

Period | 6/9/04 → 6/11/04 |

### Keywords

- Combinatorial complexity
- Fat objects
- Union of objects

### ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Computational Mathematics

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## Cite this

Aronov, B., Efrat, A., Koltun, V., & Sharir, M. (2004).

*On the union of κ-round objects in three and four dimensions*. 383-390. Paper presented at Proceedings of the Twentieth Annual Symposium on Computational Geometry (SCG'04), Brooklyn, NY, United States.