Proof Mining in R-trees and Hyperbolic Spaces

  • Authors:

  • Laurenţiu Leuştean

  • Affiliations:

  • Department of Mathematics, Darmstadt University of Technology, Schlossgartenstrasse 7, 64289 Darmstadt, Germany and Institute of Mathematics "Simion Stoilow" of the Romanian Academy, Calea Grivi&# ...

  • Venue:
  • Electronic Notes in Theoretical Computer Science (ENTCS)
  • Year:
  • 2006

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Abstract

This paper is part of the general project of proof mining, developed by Kohlenbach. By ''proof mining'' we mean the logical analysis of mathematical proofs with the aim of extracting new numerically relevant information hidden in the proofs. We present logical metatheorems for general classes of spaces from functional analysis and hyperbolic geometry, like Gromov hyperbolic spaces, R-trees and uniformly convex hyperbolic spaces (in the sense of Reich/Kirk/Kohlenbach). Our theorems are adaptations to these structures of previous metatheorems of Gerhardy and Kohlenbach, and they guarantee a-priori, under very general logical conditions, the existence of uniform bounds. We give also an application in nonlinear functional analysis, more specifically in metric fixed-point theory. Thus, we show that the uniform bound on the rate of asymptotic regularity for the Krasnoselski-Mann iterations of nonexpansive mappings in uniformly convex hyperbolic spaces obtained in a previous paper is an instance of one of our metatheorems.