Isometric embeddings of snowflakes into finite-dimensional Banach spaces
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- by Enrico Le Donne, Tapio Rajala and Erik Walsberg PDF
- Proc. Amer. Math. Soc. 146 (2018), 685-693 Request permission
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.References
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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
- MR Author ID: 867590
- 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
- MR Author ID: 838027
- 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
- MR Author ID: 1004168
- Email: erikw@illinois.edu
- 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
- © Copyright 2017 American Mathematical Society
- 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
- MathSciNet review: 3731701