### Abstract

One useful generalization of the convex hull of a set S of n points is the e{open}-strongly convex δ-hull. It is defined to be a convex polygon with vertices taken from S such that no point in S lies farther than δ outside and such that even if the vertices of are perturbed by as much as e{open}, remains convex. It was an open question as to whether an e{open}-strongly convex O(e{open})-hull existed for all positive e{open}. We give here an O(n log n) algorithm for constructing it (which thus proves its existence). This algorithm uses exact rational arithmetic. We also show how to construct an e{open}-strongly convex O(e{open} + μ)-hull in O(n log n) time using rounded arithmetic with rounding unit μ. This is the first rounded-arithmetic convex-hull algorithm which guarantees a convex output and which has error independent of n.

Original language | English (US) |
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Pages (from-to) | 345-364 |

Number of pages | 20 |

Journal | Algorithmica |

Volume | 8 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 1992 |

Externally published | Yes |

### Keywords

- Approximate arithmetic
- Convex hulls
- Exact rational arithmetic
- Numerical analysis
- Robust geometry
- Strong and weak convexity

### ASJC Scopus subject areas

- Computer Science(all)
- Computer Science Applications
- Applied Mathematics

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## Cite this

*Algorithmica*,

*8*(1), 345-364. https://doi.org/10.1007/BF01758851