### Abstract

The skeleton of a polyhedral set is the union of its edges and vertices. Let P be a set of fat, convex polytopes in three dimensions with n vertices in total, and let f
_{max} be the maximum complexity of any face of a polytope in P. We prove that the total length of the skeleton of the union of the polytopes in P is at most O(α(n)· log * n · log f
_{max}) times the sum of the skeleton lengths of the individual polytopes.

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

Pages (from-to) | 53-64 |

Number of pages | 12 |

Journal | Discrete and Computational Geometry |

Volume | 48 |

Issue number | 1 |

DOIs | |

State | Published - Jul 2012 |

### Fingerprint

### Keywords

- Combinatorial complexity
- Convex polytopes
- Fat polytopes
- Skeleton of union

### ASJC Scopus subject areas

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

### Cite this

*Discrete and Computational Geometry*,

*48*(1), 53-64. https://doi.org/10.1007/s00454-012-9422-8

**Unions of fat convex polytopes have short skeletons.** / Aronov, Boris; de Berg, Mark.

Research output: Contribution to journal › Article

*Discrete and Computational Geometry*, vol. 48, no. 1, pp. 53-64. https://doi.org/10.1007/s00454-012-9422-8

}

TY - JOUR

T1 - Unions of fat convex polytopes have short skeletons

AU - Aronov, Boris

AU - de Berg, Mark

PY - 2012/7

Y1 - 2012/7

N2 - The skeleton of a polyhedral set is the union of its edges and vertices. Let P be a set of fat, convex polytopes in three dimensions with n vertices in total, and let f max be the maximum complexity of any face of a polytope in P. We prove that the total length of the skeleton of the union of the polytopes in P is at most O(α(n)· log * n · log f max) times the sum of the skeleton lengths of the individual polytopes.

AB - The skeleton of a polyhedral set is the union of its edges and vertices. Let P be a set of fat, convex polytopes in three dimensions with n vertices in total, and let f max be the maximum complexity of any face of a polytope in P. We prove that the total length of the skeleton of the union of the polytopes in P is at most O(α(n)· log * n · log f max) times the sum of the skeleton lengths of the individual polytopes.

KW - Combinatorial complexity

KW - Convex polytopes

KW - Fat polytopes

KW - Skeleton of union

UR - http://www.scopus.com/inward/record.url?scp=84860887358&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84860887358&partnerID=8YFLogxK

U2 - 10.1007/s00454-012-9422-8

DO - 10.1007/s00454-012-9422-8

M3 - Article

VL - 48

SP - 53

EP - 64

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

SN - 0179-5376

IS - 1

ER -