  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.

Metric Geometry;Combinatorics