Abstract
We show that all the standard distances from metric geometry and functional analysis, such as Gromov—Hausdorff distance, Banach—Mazur distance, Kadets distance, Lipschitz distance, Net distance, and Hausdorff—Lipschitz distance have all the same complexity and are reducible to each other in a precisely defined way.
This is done in terms of descriptive set theory and is a part of a larger research program initiated by the authors in [8]. The paper is however targeted also to specialists in metric geometry and geometry of Banach spaces.
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Acknowledgements
M. Cúth was supported by Charles University Research program No. UNCE/SCI/023 and by the Research grant GAČR 17-04197Y. M. Doucha was supported by the GAČR project EXPRO 20-31529X, and RVO: 67985840. O. Kurka was supported by the Research grant GAČR 17-04197Y and by RVO: 67985840.
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Cúth, M., Doucha, M. & Kurka, O. Complexity of distances: Reductions of distances between metric and Banach spaces. Isr. J. Math. 248, 383–439 (2022). https://doi.org/10.1007/s11856-022-2305-7
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DOI: https://doi.org/10.1007/s11856-022-2305-7