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Double-normal pairs in the plane and on the sphere

Pach, János and Swanepoel, Konrad J. ORCID: 0000-0002-1668-887X (2015) Double-normal pairs in the plane and on the sphere. Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, 56 (2). pp. 423-438. ISSN 0138-4821

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Identification Number: 10.1007/s13366-014-0211-9


A double-normal pair of a finite set S of points from Euclidean space is a pair of points {p p,q q} from S such that S lies in the closed strip bounded by the hyperplanes through p p and q q that are perpendicular to p pq q . A double-normal pair p pq q is strict if S∖{p p,q q} lies in the open strip. We answer a question of Martini and Soltan (2006) by showing that a set of n≥3 points in the plane has at most 3⌊n/2⌋ double-normal pairs. This bound is sharp for each n≥3 . In a companion paper, we have asymptotically determined this maximum for points in R 3 . Here we show that if the set lies on some 2 -sphere, it has at most 17n/4−6 double-normal pairs. This bound is attained for infinitely many values of n . We also establish tight bounds for the maximum number of strict double-normal pairs in a set of n points in the plane and on the sphere.

Item Type: Article
Official URL:
Additional Information: © 2014 The Managing Editors
Divisions: Mathematics
Subjects: H Social Sciences > HA Statistics
JEL classification: C - Mathematical and Quantitative Methods > C6 - Mathematical Methods and Programming > C61 - Optimization Techniques; Programming Models; Dynamic Analysis
Date Deposited: 14 Jul 2014 16:19
Last Modified: 20 Sep 2021 02:05
Projects: 200021-137574, 200020-144531, OTKA NN 102029, CCF-08-30272
Funders: National Science Foundation, Swiss National Science Foundation, Hungarian Science Foundation, National service frameworks (NSFs)

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