The number of double-normals in space

Given a set V of points in , two points p, q from V form a double-normal pair, if the set V lies between two parallel hyperplanes that pass through p and q, respectively, and that are orthogonal to the segment pq. In this paper we study the maximum number of double-normal pairs in a set of n points in . It is not difficult to get from the famous ErdAs-Stone theorem that for a suitable integer and it was shown in a paper by Pach and Swanepoel that and that asymptotically . In this paper we sharpen the upper bound on k(d), which, in particular, gives and in addition to the equality established by Pach and Swanepoel. Asymptotically we get and show that this problem is connected with the problem of determining the maximum number of points in that form only acute (or non-obtuse) angles.