Double-normal pairs in space

A double-normal pair of a finite set S of points that spans Rd 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 {p,q} 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 Rd , was initiated by Martini and Soltan [Discrete Math. 290 (2005), 221–228]. It was shown in a companion paper that in the plane, this maximum is 3[n/2] , for every n > 2 . For d > 3 , it follows from the Erdős–Stone theorem in extremal graph theory that Nd(n) = 1/2(1-1/k)n2 + o(n2) for a suitable positive integer k=k(d) . Here we prove that k(3)=2 and, in general, [d/2] < d-1 . Moreover, asymptotically we have limn→∞k(d)/d=1 . The same bounds hold for the maximum number of strict double-normal pairs.