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 \(L < 1\).
The majority of this work was done while Le Roux was at the Department of Mathematics, Technische Universität Darmstadt, Germany and Pauly was at the Computer Laboratory, University of Cambridge, United Kingdom.
This project has been supported by the National Research Foundation of South Africa (NRF) and by the German Research Foundation (DFG) through the German-South African project (DFG, 445 SUA-1 13/20/0).
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Notes
- 1.
Which was put to the authors by Kohlenbach.
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Brattka, V., Le Roux, S., Miller, J.S., Pauly, A. (2016). The Brouwer Fixed Point Theorem Revisited. In: Beckmann, A., Bienvenu, L., Jonoska, N. (eds) Pursuit of the Universal. CiE 2016. Lecture Notes in Computer Science(), vol 9709. Springer, Cham. https://doi.org/10.1007/978-3-319-40189-8_6
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