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arXiv · 1608.07342

Bisectors determining unique pairs of points in the bidisk

Virginie Charette·Todd A. Drumm·Youngju Kim
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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 $\mathbb{H}^2 \times\mathbb{H}^2$.

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Virginie Charette, Todd A. Drumm, Youngju Kim. 2016-08-26. Bisectors determining unique pairs of points in the bidisk. https://arxiv.org/abs/1608.07342

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