Double-normal pairs in space

A double-normal pair of a finite set S of points from R 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 pq 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 R, was initiated by Martini and Soltan (2006). It was shown in a companion paper that in the plane, this maximum is 3bn/2c, for every n > 2. For d ≥ 3, it follows from the Erdős-Stone theorem in extremal graph theory that Nd(n) = 12 (1− 1/k)n 2 + o(n) for a suitable positive integer k = k(d). Here we prove that k(3) = 2 and, in general, dd/2e ≤ k(d) ≤ d − 1. Moreover, asymptotically we have limn→∞ k(d)/d = 1. The same bounds hold for the maximum number of strict double-normal pairs.