### 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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Title of host publication | Proceedings of the Twentieth Annual Symposium on Computational Geometry (SCG'04) |

Pages | 383-390 |

Number of pages | 8 |

State | Published - 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) |
---|---|

Country | United States |

City | Brooklyn, NY |

Period | 6/9/04 → 6/11/04 |

### Fingerprint

### Keywords

- Combinatorial complexity
- Fat objects
- Union of objects

### ASJC Scopus subject areas

- Software
- Geometry and Topology
- Safety, Risk, Reliability and Quality
- Chemical Health and Safety

### Cite this

*Proceedings of the Twentieth Annual Symposium on Computational Geometry (SCG'04)*(pp. 383-390)

**On the union of κ-round objects in three and four dimensions.** / Aronov, Boris; Efrat, Alon; Koltun, Vladlen; Sharir, Micha.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Proceedings of the Twentieth Annual Symposium on Computational Geometry (SCG'04).*pp. 383-390, Proceedings of the Twentieth Annual Symposium on Computational Geometry (SCG'04), Brooklyn, NY, United States, 6/9/04.

}

TY - GEN

T1 - On the union of κ-round objects in three and four dimensions

AU - Aronov, Boris

AU - Efrat, Alon

AU - Koltun, Vladlen

AU - Sharir, Micha

PY - 2004

Y1 - 2004

N2 - A compact body 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.

AB - A compact body 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.

KW - Combinatorial complexity

KW - Fat objects

KW - Union of objects

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M3 - Conference contribution

SP - 383

EP - 390

BT - Proceedings of the Twentieth Annual Symposium on Computational Geometry (SCG'04)

ER -