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Isometric embeddings of snowflakes into finite-dimensional Banach spaces


Authors: Enrico Le Donne, Tapio Rajala and Erik Walsberg
Journal: Proc. Amer. Math. Soc. 146 (2018), 685-693
MSC (2010): Primary 30L05, 46B85, 54C25, 54E40, 28A80
DOI: https://doi.org/10.1090/proc/13778
Published electronically: October 12, 2017
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Abstract: We consider a general notion of snowflake of a metric space by composing the distance with a nontrivial concave function. We prove that a snowflake of a metric space $ X$ isometrically embeds into some finite-dimensional normed space if and only if $ X$ is finite. In the case of power functions we give a uniform bound on the cardinality of $ X$ depending only on the power exponent and the dimension of the vector space.


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Additional Information

Enrico Le Donne
Affiliation: Department of Mathematics and Statistics, University of Jyvaskyla, P.O. Box 35 (MaD), FI-40014 University of Jyvaskyla, Finland
Email: enrico.e.ledonne@jyu.fi

Tapio Rajala
Affiliation: Department of Mathematics and Statistics, University of Jyvaskyla, P.O. Box 35 (MaD), FI-40014 University of Jyvaskyla, Finland
Email: tapio.m.rajala@jyu.fi

Erik Walsberg
Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, Illinois 61802
Email: erikw@illinois.edu

DOI: https://doi.org/10.1090/proc/13778
Received by editor(s): November 23, 2016
Received by editor(s) in revised form: February 10, 2017, and March 27, 2017
Published electronically: October 12, 2017
Additional Notes: The first and second authors acknowledge the support of the Academy of Finland, projects no. 288501 and 274372
The third author acknowledges the support of the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC Grant agreement no. 291111/ MODAG
Communicated by: Jeremy Tyson
Article copyright: © Copyright 2017 American Mathematical Society

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