Coarse higher medians, symmetric spaces, and convex projective geometry

Abstract

We introduce a notion of coarse $r$-median spaces that is a higher-rank analog of Bowditch's coarse median spaces. Our notion is stable under quasi-isometries and recovers Bowditch's coarse medians when $r$ equals 1. We prove that several families of higher rank-symmetric spaces admit coarse $r$-medians with $r$ being the rank of the symmetric space. In particular, our list includes plenty of examples that are known to not admit Bowditch's coarse medians. Our main tools come from convex projective geometry, and we prove the existence of coarse higher medians on all divisible as well as quasi-homogeneous properly convex domains.