Pach, János and Swanepoel, Konrad J.
(2015)
*Double-normal pairs in space.*
Mathematika, 61 (1).
pp. 259-272.
ISSN 0025-5793

## Abstract

A double-normal pair of a finite set S of points that spans Rd is a pair of points {p,q} from S such that S lies in the closed strip bounded by the hyperplanes through p and q perpendicular to pq . A double-normal pair {p,q} is strict if S\{p,q} lies in the open strip. The problem of estimating the maximum number Nd(n) of double-normal pairs in a set of n points in Rd , was initiated by Martini and Soltan [Discrete Math. 290 (2005), 221–228]. It was shown in a companion paper that in the plane, this maximum is 3[n/2] , for every n > 2 . For d > 3 , it follows from the Erdős–Stone theorem in extremal graph theory that Nd(n) = 1/2(1-1/k)n2 + o(n2) for a suitable positive integer k=k(d) . Here we prove that k(3)=2 and, in general, [d/2] < d-1 . Moreover, asymptotically we have limn→∞k(d)/d=1 . The same bounds hold for the maximum number of strict double-normal pairs.

Item Type: | Article |
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Official URL: | http://journals.cambridge.org/ |

Additional Information: | © 2015 University College London |

Divisions: | Management |

Subjects: | Q Science > QA Mathematics |

Sets: | Departments > Management |

Date Deposited: | 05 Sep 2014 08:38 |

Last Modified: | 20 Mar 2019 02:37 |

Projects: | 200021-137574, 200020-144531, OTKA NN 102029 under the EuroGIGA programs ComPoSe and GraDR, CCF-08-30272 |

Funders: | Swiss National Science Foundation, Swiss National Science Foundation, Hungarian Science Foundation, National Science Foundation (NSF) |

URI: | http://eprints.lse.ac.uk/id/eprint/59275 |

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