The number of double-normals in space

Abstract

Given a set V of points in ${\mathbb {R}}^d$, 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 $N_d(n)$ of double-normal pairs in a set of n points in ${\mathbb {R}}^d$. It is not difficult to get from the famous Erdős–Stone theorem that $N_d(n) = \frac{1}{2}(1-1/k)n^2+o(n^2)$ for a suitable integer $k = k(d)$ and it was shown in a paper by Pach and Swanepoel that $\lceil d/2\rceil \le k(d)\le d-1$ and that asymptotically $k(d)\gtrsim d-O(\log d)$. In this paper we sharpen the upper bound on k(d), which, in particular, gives $k(4)=2$ and $k(5)=3$ in addition to the equality $k(3)=2$ established by Pach and Swanepoel. Asymptotically we get $k(d)\le d- \log _2k(d) = d - (1+ o(1)) \log _2k(d)$ and show that this problem is connected with the problem of determining the maximum number of points in ${\mathbb {R}}^d$ that form only acute (or non-obtuse) angles.