Abstract
We prove that the regular \({n\times n}\) square grid of points in the integer lattice \({\mathbb{Z}^{2}}\) cannot be recovered from an arbitrary \({n^{2}}\)-element subset of \({\mathbb{Z}^{2}}\) via a mapping with prescribed Lipschitz constant (independent of n). This answers negatively a question of Feige from 2002. Our resolution of Feige’s question takes place largely in a continuous setting and is based on some new results for Lipschitz mappings falling into two broad areas of interest, which we study independently. Firstly the present work contains a detailed investigation of Lipschitz regular mappings on Euclidean spaces, with emphasis on their bilipschitz decomposability in a sense comparable to that of the well known result of Jones. Secondly, we build on work of Burago and Kleiner and McMullen on non-realisable densities. We verify the existence, and further prevalence, of strongly non-realisable densities inside spaces of continuous functions.
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Acknowledgments
Open access funding provided by University of Innsbruck and Medical University of Innsbruck. The authors would like to thank Uriel Feige for posing the question on which this work is based and for providing them with details of its motivation. Moreover, they would like to thank Florian Baumgartner for fruitful discussions during the early stages of the presented work and Martin Tancer for his helpful remarks regarding the manuscript. Furthermore, they would like to thank Guy C. David for pointing out the reference to the article [BK02] of Bonk and Kleiner. Last but not least, the authors thank the anonymous referees for careful reading and many helpful suggestions that improved the article.
The authors wish to place on record their gratitude to Jirka Matoušek, whose deep insight into the problem gave us the right direction from the start of the presented research. The second named author would also like to express his gratitude to Jirka Matoušek for valuable guidance during his undergraduate studies.
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V.K. was partially supported by the project GAČR 16-01602Y. He was also supported by the project FWF P23628-N18 during his stays at Universit¨at Innsbruck. E.K. was partially supported by the project FWF P23628-N18.
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Dymond, M., Kaluža, V. & Kopecká, E. Mapping n Grid Points Onto a Square Forces an Arbitrarily Large Lipschitz Constant. Geom. Funct. Anal. 28, 589–644 (2018). https://doi.org/10.1007/s00039-018-0445-z
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DOI: https://doi.org/10.1007/s00039-018-0445-z