The Brouwer Fixed Point Theorem Revisited

  • Vasco Brattka
  • Stéphane Le Roux
  • Joseph S. Miller
  • Arno Pauly
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9709)

Abstract

We revisit the investigation of the computational content of the Brouwer Fixed Point Theorem in [7], and answer the two open questions from that work. First, we show that the computational hardness is independent of the dimension, as long as it is greater than 1 (in [7] this was only established for dimension greater than 2). Second, we show that restricting the Brouwer Fixed Point Theorem to L-Lipschitz functions for any \(L > 1\) also does not change the computational strength, which together with prior results establishes a trichotomy for \(L > 1\), \(L = 1\) and \(L < 1\).

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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Vasco Brattka
    • 1
    • 2
  • Stéphane Le Roux
    • 3
  • Joseph S. Miller
    • 4
  • Arno Pauly
    • 3
  1. 1.Faculty of Computer ScienceUniversity of the Armed Forces MunichNeubibergGermany
  2. 2.Department of Mathematics and Applied MathematicsUniversity of Cape TownCape TownSouth Africa
  3. 3.Département d’InformatiqueUniversité libre de BruxellesBrusselsBelgium
  4. 4.Department of MathematicsUniversity of Wisconsin–MadisonMadisonUSA

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