Double-normal pairs in space

Abstract

A double-normal pair of a finite set $S$ of points from $R^d$ 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 $S\setminus\{p,q\}$ lies in the open strip. The problem of estimating the maximum number $N_d(n)$ of double-normal pairs in a set of $n$ points in $R^d$, was initiated by Martini and Soltan (2006). It was shown in a companion paper that in the plane, this maximum is $3\lfloor n/2\rfloor$, for every $n>2$. For $d\geq 3$, it follows from the Erd\H{o}s-Stone theorem in extremal graph theory that $N_d(n)=\frac12(1-1/k)n^2 + o(n^2)$ for a suitable positive integer $k=k(d)$. Here we prove that $k(3)=2$ and, in general, $\lceil d/2\rceil \leq k(d)\leq d-1$. Moreover, asymptotically we have $\lim_{n\rightarrow\infty}k(d)/d=1$. The same bounds hold for the maximum number of strict double-normal pairs.