Ptolemy spaces

Some on-line resources

• I. J. Schoenberg, A Remark on M. M. Day's Characterization of Inner-Product Spaces and a Conjecture of L. M. Blumenthal,.
  [A real normed vector space, which is ptolemaic, is an inner product space]


• D.C.Kay, Ptolemaic metric spaces and the characterization of geodesics by vanishing metric curvature, Ph.D. thesis, Michigan State University,.
  [A Riemannian locally ptolemaic space is nonpositively curved.]


• D.C.Kay, The Ptolemaic inequality in Hilbert geometries.
  [A Hilbert geometry is locally ptolemaic if and only if it is hyperbolic.]


• T. Foertsch, V. Schroeder, Hyperbolicity, CAT(−1)-spaces and the Ptolemy Inequality.
  [All Bourdon and Hamenstaedt metrics on ∂_∞Y , where Y is CAT(−1), generate a Ptolemy space.]


• T. Foertsch, A. Lytchak, V. Schroeder, Non-positive curvature and the Ptolemy inequality.
  [A geodesic metric space is CAT(0) if and only if it is ptolemaic and Busemann convex, a ptolemaic proper geodesic metric space is uniquely geodesic.]


• S. Buckley, K. Falk, D. Wraith, Ptolemaic spaces and CAT(0).
  [A complete Riemannian manifold is ptolemaic if and only if it is a Hadamard manifold, a Finsler ptolemaic manifold is Riemannian.]


• S. Buyalo, V. Schroeder, Möbius structures and Ptolemy spaces: boundary at infinity of complex hyperbolic spaces.
  [Conjecture that every compact Ptolemy space with circles and many space inversions is M&"obius equivalent to the boundary at infinity of a rank one symmetric space of noncompact type. (proved for the class of complex hyperbolic spaces).]


• V. Adiyasuren, T.S. Batbold, Some Inequalities in Ptolemaic Spaces.
  [Several inequalities in Ptolemaic Spaces.]


• A. A. Dovgoshei, E. A. Petrov, Ptolemaic Spaces .
  [Extremally non-Ptolemaic metric spaces, four-point pseudometric spaces that are maximally Ptolemaic.]


• R. Miao Hyperbolic Spaces and Ptolemy Möbius Structures, PhD Thesis, Zuerich.
  [Studies the relationship between the CAT(κ) spaces and PTκ spaces.]


• Z. Zhang, Y. Xiao, Strongly hyperbolic metrics on Ptolemy spaces.
  [The log-metric on the Ptolemy space is a strongly hyperbolic metric.]


• I. Platis, Cross-ratios and the Ptolemaean inequality in boundaries of symmetric spaces of rank 1.
  [The Ptolemaean inequality and the Theorem of Ptolemaeus hold in the setting of the boundary of symmetric Riemannian spaces of rank 1 and of negative curvature.]


• M. Gomez, F. Memoli, The Four Point Condition: An Elementary Tropicalization of Ptolemy’s Inequality .
  [A limit of a family of Ptolemaic inequalities in CAT-spaces.]