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K-Euler’s inequality in Aleksandrov’s CAT(K) space

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Abstract

We establish the K-Euler’s inequality and the extremal theorem for equality in the K-Euler’s inequality in Aleksandrov’s \({\text {CAT}}\left( K\right) \) space for non-zero K.

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References

  1. Alexandrow, A.D.: Über eine Verallgemeinerung der Riemannschen Geometrie. Schr. Forschungsinst. Math. 1, 33–84 (1957)

    MATH  MathSciNet  Google Scholar 

  2. Berg, I.D., Nikolaev, I.G.: Quasilinearization and curvature of Aleksandrov spaces. Geom. Dedicata. 133, 195–218 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bridson, M.R., Haefliger, A.: Metric Spaces of Non-positive Curvature. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319. Springer, Berlin (1999)

  4. Enflo, P.: On the nonexistence of uniform homeomorphisms between \(L_p\)-spaces. Ark. Mat. 8, 103–105 (1969)

  5. Euler, L.: Variae demonstrationes geometriae. Novi Commentarii academiae scientiarum Petropolitanae 1, 49–66 (1750) (Opera Omnia, Ser. 1, 26, 29–32 (1953)

  6. Lafont, J.-F., Prassidis, S.: Roundness properties of groups. Geom. Dedicata. 117, 137–160 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  7. Reshetnyak, Y.G.: Non-expanding mappings in a space of curvature not greater than \(K\). Sibirsk. Mat. Zh. 9, 918–927 (1968). (in Russian), English translation: Sib. Math. J. 9, 683–689 (1968)

  8. Sato, T.: An alternative proof of Berg and Nikolaev’s characterization of \(\operatorname{CAT}\left(0\right)\)-spaces via quadrilateral inequality. Arch. Math. 93(5), 487–490 (2009)

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Correspondence to Igor G. Nikolaev.

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Berg, I.D., Nikolaev, I.G. K-Euler’s inequality in Aleksandrov’s CAT(K) space. J. Geom. 108, 869–878 (2017). https://doi.org/10.1007/s00022-017-0381-3

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  • DOI: https://doi.org/10.1007/s00022-017-0381-3

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