TITLE: Bisectors in symmetric spaces
QUESTION [5 upvotes]: In William Goldman's book Complex Hyperbolic Geometry, bisector hypersurfaces play an important role. Given two points $x,y$, the bisector is the set of points equidistant from $x$ and from $y$. Do they also play an important role in the study of higher rank symmetric spaces, and do they admit concrete descriptions, say in the Siegel upper half-plane? References are especially welcome.

REPLY [3 votes]: Although I cannot address your specific question, of whether bisectors "play an important role in the study of higher rank symmetric spaces," 
I can say that
bisectors are the essence of Voronoi diagrams.
So the 
2009 paper by
Frank Nielsen and Richard Nock, entitled
"Hyperbolic Voronoi diagrams made easy"
(arXiv link), could well be relevant:

We present a simple framework to compute hyperbolic Voronoi diagrams of finite point sets as affine diagrams. We prove that bisectors in Klein's non-conformal disk model are hyperplanes that can be interpreted as power bisectors of Euclidean balls. [...]