Double-normal pairs in the plane and on the sphere

A double-normal pair of a finite set S of points from Euclidean space 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 that are perpendicular to pq. A double-normal pair pq is strict if S \ {p, 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 3bn/2c 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. 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.