Double-normal pairs in the plane and on the sphere

Abstract

A double-normal pair of a finite set $S$ of points from Euclidean space is a pair of points $\left\{ \varvec{p},\varvec{q}\right\}$ from $S$ such that $S$ lies in the closed strip bounded by the hyperplanes through $\varvec{p}$ and $\varvec{q}$ that are perpendicular to $\varvec{p}\varvec{q}$. A double-normal pair $\varvec{p}\varvec{q}$ is strict if $S{\setminus }\left\{ \varvec{p},\varvec{q}\right\}$ 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 $3\lfloor n/2\rfloor$ 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 $\mathbb {R}^3$. Here we show that if the set lies on some $2$-sphere, it has at most $17n/4 - 6$ 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.