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Lipschitz-free Spaces on Finite Metric Spaces

Part of: General convexity Normed linear spaces and Banach spaces; Banach lattices Analysis on metric spaces Nontrigonometric harmonic analysis

Published online by Cambridge University Press:  13 February 2019

Stephen J. Dilworth ,
Denka Kutzarova  and
Mikhail I. Ostrovskii
Stephen J. Dilworth
Affiliation:
Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA Email: dilworth@math.sc.edu
Denka Kutzarova
Affiliation:
Department of Mathematics University of Illinois at Urbana-Champaign Urbana, IL 61801, USA Institute of Mathematics and Informatics, Bulgarian Academy of Sciences Email: denka@math.uiuc.edu
Mikhail I. Ostrovskii
Affiliation:
Department of Mathematics and Computer Science, St. John’s University, 8000 Utopia Parkway, Queens, NY 11439, USA Email: ostrovsm@stjohns.edu
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Abstract

Main results of the paper are as follows:

(1) For any finite metric space $M$ the Lipschitz-free space on $M$ contains a large well-complemented subspace that is close to $\ell _{1}^{n}$.

(2) Lipschitz-free spaces on large classes of recursively defined sequences of graphs are not uniformly isomorphic to $\ell _{1}^{n}$ of the corresponding dimensions. These classes contain well-known families of diamond graphs and Laakso graphs.

Interesting features of our approach are: (a) We consider averages over groups of cycle-preserving bijections of edge sets of graphs that are not necessarily graph automorphisms. (b) In the case of such recursive families of graphs as Laakso graphs, we use the well-known approach of Grünbaum (1960) and Rudin (1962) for estimating projection constants in the case where invariant projections are not unique.

Keywords

Arens–Eells spacediamond graphearth mover distanceKantorovich–Rubinstein distanceLaakso graphLipschitz-free spacerecursive family of graphstransportation costWasserstein distance

MSC classification

Primary: 52A21: Finite-dimensional Banach spaces (including special norms, zonoids, etc.)
Secondary: 30L05: Geometric embeddings of metric spaces 42C10: Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.) 46B07: Local theory of Banach spaces 46B20: Geometry and structure of normed linear spaces 46B85: Embeddings of discrete metric spaces into Banach spaces; applications in topology and computer science
Type
Article
Information
Canadian Journal of Mathematics , Volume 72 , Issue 3 , June 2020 , pp. 774 - 804
Copyright
© Canadian Mathematical Society 2019

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Footnotes

Author S. D. was supported by the National Science Foundation under Grant Number DMS–1361461. Authors S. D. and D. K. were supported by the Workshop in Analysis and Probability at Texas A&M University in 2017. Author M. O. was supported by the National Science Foundation under Grant Number DMS–1700176.

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