### Abstract

A compact set 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 in the worst case.

Original language | English (US) |
---|---|

Pages (from-to) | 511-526 |

Number of pages | 16 |

Journal | Discrete and Computational Geometry |

Volume | 36 |

Issue number | 4 |

DOIs | |

State | Published - 2006 |

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### ASJC Scopus subject areas

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

### Cite this

*Discrete and Computational Geometry*,

*36*(4), 511-526. https://doi.org/10.1007/s00454-006-1263-x

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

Research output: Contribution to journal › Article

*Discrete and Computational Geometry*, vol. 36, no. 4, pp. 511-526. https://doi.org/10.1007/s00454-006-1263-x

}

TY - JOUR

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 - 2006

Y1 - 2006

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=33750314098&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33750314098&partnerID=8YFLogxK

U2 - 10.1007/s00454-006-1263-x

DO - 10.1007/s00454-006-1263-x

M3 - Article

AN - SCOPUS:33750314098

VL - 36

SP - 511

EP - 526

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

SN - 0179-5376

IS - 4

ER -