Quasilinearization and curvature of Aleksandrov spaces

I. D. Berg, I. G. Nikolaev

Research output: Contribution to journalArticlepeer-review


We present a new distance characterization of Aleksandrov spaces of non-positive curvature. By introducing a quasilinearization for abstract metric spaces we draw an analogy between characterization of Aleksandrov spaces and inner product spaces; the quasi-inner product is defined by means of the quadrilateral cosine-a metric substitute for the angular measure between two directions at different points. Our main result states that a geodesically connected metric space (M,ρ) is an Aleksandrov R0 domain (also known as a CAT(0) space) if and only if the quadrilateral cosine does not exceed one for every two pairs of distinct points in M. We also observe that a geodesically connected metric space (M, ρ) is an R0 domain if and only if, for every quadruple of points in M, the quadrilateral inequality (known as Euler's inequality in 2) holds. As a corollary of our main result we give necessary and sufficient conditions for a semimetric space to be an R0domain. Our results provide a complete solution to the Curvature Problem posed by Gromov in the context of metric spaces of non-positive curvature.

Original languageEnglish (US)
Pages (from-to)195-218
Number of pages24
JournalGeometriae Dedicata
Issue number1
StatePublished - Apr 2008


  • 2-Roundness
  • Aleksandrov space
  • Quadrilateral cosine
  • Quadrilateral inequality
  • ℛ domain

ASJC Scopus subject areas

  • Geometry and Topology


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