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
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.
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