  Bisectors are equidistant hypersurfaces between two points and are basic
objects in a metric geometry. They play an important part in understanding the
action of subgroups of isometries on a metric space. In many metric geometries
(spherical, Euclidean, hyperbolic, complex hyperbolic, to name a few) bisectors
do not uniquely determine a pair of points, in the following sense\,:
completely different sets of points share a common bisector. The above examples
of this non-uniqueness are all rank $1$ symmetric spaces.
  However, as we show in this paper, bisectors in the usual $L^2$ metric are
such for a unique pair of points in the rank $2$ geometry $\mathbb{H}^2
\times\mathbb{H}^2$.
