Quasilinearization and curvature of Aleksandrov spaces |
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Authors: | I D Berg I G Nikolaev |
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Institution: | (1) University of Illinois at Urbana-Champaign, 273 Altgeld Hall, 1409 W. Green St., Urbana, IL 61801, USA |
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Abstract: | 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 is an Aleksandrov 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 . We also observe that a geodesically connected metric space is an domain if and only if, for every quadruple of points in , the quadrilateral inequality (known as Euler’s inequality in ) holds. As a corollary of our main result we give necessary and sufficient conditions for a semimetric space to be an domain. Our results provide a complete solution to the Curvature Problem posed by Gromov in the context of metric spaces
of non-positive curvature.
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Keywords: | Aleksandrov space domain" target="_blank">gif" alt="$${\Re_{0}}$$" align="middle" border="0"> domain Quadrilateral cosine Quadrilateral inequality 2-Roundness |
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