The Tits Metric and Visibility Spaces
Let X be a complete CAT(0) space. In the preceding chapter we constructed a boundary at infinity ∂X and studied the cone topology on it. (This topology makes ∂X compact if X is proper.) ∂ℍ n and ∂E n are homeomorphic in the cone topology, but at an intuitive level they appear quite different when viewed from within the space. Consider how the apparent distance between two points at infinity changes as one moves around in E n and ℍ n : in E n the angle subtended at the eye of an observer by the geodesic rays going to two fixed points at infinity does not depend on where the observer is standing; in ℍ n the angle depends very much on where the observer is standing, and by standing in the right place he can make the angle π. Thus by recording the view of ∂X from various points inside X one obtains a metric structure that discriminates between the boundaries of E n and ℍ n . In this chapter we shall consider the same metric structure in the context of complete CAT(0) spaces.
KeywordsConvex Hull Distinct Point Visibility Space Euclidean Plane Geodesic Segment
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