Bisectors determining unique pairs of points in the bidisk

TL;DR

This paper investigates the uniqueness of point pairs determined by bisectors in various metric geometries, revealing that in the rank 2 geometry ^2, bisectors uniquely identify pairs of points, unlike in rank 1 spaces.

Contribution

The paper demonstrates that in the ^2 geometry, bisectors uniquely determine pairs of points, contrasting with non-uniqueness in rank 1 symmetric spaces.

Findings

Bisectors in ^2 uniquely determine pairs of points.

Non-uniqueness of bisectors occurs in rank 1 symmetric spaces.

In ^2, bisectors serve as unique identifiers for point pairs.

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 $1$ 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 $\mathbb{H}^2 \times\mathbb{H}^2$.