Pach, János and Swanepoel, Konrad J. ORCID: 000000021668887X (2015) Doublenormal pairs in the plane and on the sphere. Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, 56 (2). pp. 423438. ISSN 01384821

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Abstract
A doublenormal 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 doublenormal 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⌋ doublenormal 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 doublenormal pairs. This bound is attained for infinitely many values of n . We also establish tight bounds for the maximum number of strict doublenormal pairs in a set of n points in the plane and on the sphere.
Item Type:  Article 

Official URL:  http://www.springer.com/mathematics/algebra/journa... 
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 
Sets:  Departments > Mathematics Collections > Economists Online 
Date Deposited:  14 Jul 2014 16:19 
Last Modified:  20 Sep 2021 02:05 
Projects:  200021137574, 200020144531, OTKA NN 102029, CCF0830272 
Funders:  National Science Foundation, Swiss National Science Foundation, Hungarian Science Foundation, National service frameworks (NSFs) 
URI:  http://eprints.lse.ac.uk/id/eprint/57687 
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