Double-normal pairs in the plane and on the sphere

A double-normal pair of a finite set $S$ of points from Euclidean space is a pair of points $\{p,q\}$ from $S$ such that $S$ lies in the closed strip bounded by the hyperplanes through $p$ and $q$ that are perpendicular to $pq$. A double-normal pair $pq$ is strict if $S\setminus \{p,q\}$ lies in the open strip. We answer a question of Martini and Soltan (2006) by showing that a set of $n\ge 3$ points in the plane has at most $3\lfloor n\mathrm{/}2\rfloor$ double-normal pairs. This bound is sharp for each $n\ge 3$. In a companion paper, we have asymptotically determined this maximum for points in $R^3$. Here we show that if the set lies on some 2-sphere, it has at most $17n\mathrm{/}4-6$ double-normal pairs. This bound is attained for infinitely many values of $n$. We also establish tight bounds for the maximum number of strict double-normal pairs in a set of $n$ points in the plane and on the sphere.