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A Proof-Theoretic Bound Extraction Theorem for CAT\((\kappa )\)-Spaces

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Abstract

Starting in 2005, general logical metatheorems have been developed that guarantee the extractability of uniform effective bounds from large classes of proofs of theorems that involve abstract metric structures X. In this paper we adapt this to the class of CAT\((\kappa \))-spaces X for \(\kappa >0\) and establish a new metatheorem that explains specific bound extractions that recently have been achieved in this context as instances of a general logical phenomenon.

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References

  1. Ariza-Ruiz, D., A. Fernández-León, G. López-Acedo, and A. Nicolae, Chebyshev sets in geodesic spaces, Journal of Approximation Theory 207:265–282, 2016.

    Article  Google Scholar 

  2. Berg, I. D., and I. G. Nikolaev, A \(K\)-quadrilateral cosine characterization of Aleksandrov spaces of curvature bounded above. arXiv:1512.01736v1, 2015.

  3. Bridson, M. R., and A. Haefliger, Metric Spaces of Non-positive Curvature, Springer, Berlin, 1999.

    Book  Google Scholar 

  4. Cho, S., A variant of continuous logic and applications to fixed point theory. arXiv:1610.05397, 2016.

  5. Espínola, R., and A. Fernández-León, CAT\((\kappa )\)-spaces, weak convergence and fixed points. Journal of Mathematical Analysis and Applications 353:410–427, 2009.

    Article  Google Scholar 

  6. Gerhardy, P., and U. Kohlenbach, General logical metatheorems for functional analysis, Transactions of American Mathematical Society 360:2615–2660, 2008.

    Article  Google Scholar 

  7. Günzel, D., and U. Kohlenbach, Logical metatheorems for abstract spaces axiomatized in positive bounded logic, Advances in Mathematics 290:503–551, 2016.

    Article  Google Scholar 

  8. Kohlenbach, U., Effective moduli from ineffective uniqueness proofs. An unwinding of de La Vallée Poussin’s proof for Chebycheff approximation, Annals of Pure and Applied Logic 64:27–94, 1993.

    Article  Google Scholar 

  9. Kohlenbach, U., Proof Theory and Computational Analysis, Electronic Notes in Theoretical Computer Science, vol. 13, Elsevier, Amsterdam, 1998, 34 pp.

  10. Kohlenbach, U., Some logical metatheorems with applications in functional analysis, Transactions of the American Mathematical Society 357(1):89–128, 2005.

    Article  Google Scholar 

  11. Kohlenbach, U., Applied Proof Theory: Proof Interpretations and Their Use in Mathematics, Springer Monographs in Mathematics, Springer, Berlin, 2008, 536 pp.

  12. Kohlenbach, U., On the quantitative asymptotic behavior of strongly nonexpansive mappings in Banach and geodesic spaces, Israel Journal of Mathematics 216:215–246, 2016.

    Article  Google Scholar 

  13. Kohlenbach, U., Recent progress in proof mining in nonlinear analysis, To appear in forthcoming book with invited papers by recipients of the Gödel Centenary Research Prize Fellowship.

  14. Leuştean, L., and A. Nicolae, Effective results on nonlinear ergodic averages in CAT\((\kappa )\) spaces, Ergodic Theory and Dynamical Systems 36:2580–2601, 2016.

    Article  Google Scholar 

  15. Pia̧tek, B., Halpern iteration in CAT(\(\kappa \)) spaces, Acta Mathematica Sinica, English Series 27:635–646, 2011.

    Article  Google Scholar 

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Correspondence to U. Kohlenbach.

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presented by Daniele Mundici

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Kohlenbach, U., Nicolae, A. A Proof-Theoretic Bound Extraction Theorem for CAT\((\kappa )\)-Spaces. Stud Logica 105, 611–624 (2017). https://doi.org/10.1007/s11225-016-9702-z

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  • DOI: https://doi.org/10.1007/s11225-016-9702-z

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