Number of double-normal pairs 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 Erd\H{o}s-Stone theorem that $N_d(n) = \frac 12(1-1/k)n^2+o(n^2)$ for a suitable integer $k = k(d)$ and it was shown in the paper by J. Pach and K. 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 J. Pach and K. 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 pairwise acute (or non-obtuse) angles.