On a distance characterization of A.D. Aleksandrov spaces of nonpositive curvature

I. D. Berg, I. G. Nikolaev

Research output: Contribution to journalArticlepeer-review

Abstract

A new distance characterization of A.D. Aleksandrov domains of a space of nonpositive curvature has been provided. Aleksandrov domains were characterized by introducing a quasilinearization for an arbitrary metric space and formulating a Cauchy-Schwarz inequality, which implies an upper curvature bound of zero. A semimetric space with a quasi-inner product on it is called the quasilinearization of the semimetric space. It was proved that a semimetric space is isometric to a complete Aleksandrov domain if and only if it is weakly convex, each Cauchy sequence in this semimetric space has a limit, and it satisfies the four point cosq condition. It was also proved that a geodesically connected metric space is an Aleksandrov domain if and only if it satisfies the quadrilateral inequality condition.

Original languageEnglish (US)
Pages (from-to)336-338
Number of pages3
JournalDoklady Mathematics
Volume75
Issue number3
DOIs
StatePublished - Jun 2007

ASJC Scopus subject areas

  • Mathematics(all)

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