  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 $n\geq 3$ points in the plane has at most $3\lfloor n/2\rfloor$
double-normal pairs. This bound is sharp for each $n\geq 3$.
  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 $17n/4 - 6$ 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.
