Bisectors determining unique pairs of points in the bidisk

Abstract

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 [Formula: see text] symmetric spaces. However, generically, bisectors in the usual [Formula: see text] metric are such for a unique pair of points in the rank [Formula: see text] geometry [Formula: see text]. This result indicates the striking assertion that non-uniqueness of bisectors holds for “most” geometries.