Hyperbolicity, CAT(−1)-spaces and the Ptolemy inequality

Abstract

Using a four points inequality for the boundary of CAT(−1)-spaces we study the relation between Gromov hyperbolic spaces and CAT(−1)-spaces.

Best constants for metric space inversion inequalities

“…A natural first question related to sharpness is therefore to investigate the sharpness of the first inequality in (1) for general metric spaces. Note that the second inequality is sharp since = I whenever X is a CAT(0) space, as explained in [5] or [2]; see also [6] and [4] for more on Ptolemaic spaces (i.e. spaces in which = I for all ).…”

Incidence Axioms for the Boundary at Infinity of Complex Hyperbolic Spaces

“…Given a complex hyperbolic plane E ⊂ M and a point a ∈ E, the orthogonal complement E ⊥ ⊂ M to E at a is a totally geodesic subspace isometric to C H , except for the case ξ = ξ = ω, is an extended (unbounded) metric on Y with in nitely remote point ω, where (ξ |ξ ) b is the Gromov product with respect to b, see [3, sect.3.4.2]. Since M is a CAT(− )-space, the metrics da, d b satisfy the Ptolemy inequality and furthermore all these metrics are pairwise Möbius equivalent, see [6].…”

“…any metric d ∈ M is Möbius equivalent to some metric da, a ∈ M. Then Y endowed with M is a compact Ptolemy space. Every extended metric d ∈ M is of type d = d b for some Busemann function b : M → R, while a bounded metric d ∈ M does not necessary coincide with λda, for some a ∈ M and λ > , see [6]. We emphasize that metrics of M are neither Carnot-Carathéodory metrics nor length metrics.…”

Ptolemaic Spaces and Cat(0)

“…In fact, this follows readily from the fact that i p is the pullback of Euclidean distance under inversion in the unit sphere. Ptolemy's inequality has also been studied in more general settings than Euclidean space (see for instance [1,10,14]). Of particular interest to us is the result of Schoenberg [15] which says that a normed space is Ptolemaic if and only if its norm is induced by an inner product.…”