Abstract
We give a simple example of a countable metric graph M such that M Lipschitz embeds with distortion strictly less than 2 into a Banach space X only if X contains an isomorphic copy of l 1. Further we show that, for each ordinal α < ω 1, the space C([0, ω α]) does not Lipschitz embed into C(K) with distortion strictly less than 2 unless K (α) ≠ 0. Also \(C\left( {\left[ {0,{\omega ^{{\omega ^\alpha }}}} \right]} \right)\) does not Lipschitz embed into a Banach space X with distortion strictly less than 2 unless Sz(X) ≥ ω α+1.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
I. Aharoni, Every separable metric space is Lipschitz equivalent to a subset of c+ 0, Israel Journal of Mathematics 19 (1974), 284–291.
F. Albiac and N. Kalton, Topics in Banach Space Theory, Graduate Texts in Mathematics, Vol. 233, Springer, New York, 2006.
D. Amir, On isomorphisms of continuous function spaces, Israel Journal of Mathematics 3 (1965), 205–210.
F. Baudier, A topological obstruction for small-distortion embeddability into spaces of continuous functions on countable compact metric spaces, arXiv:1305.4025[math.FA].
F. Baudier, D. Freeman, T. Schlumprecht and A. Zsák, The metric geometry of the Hamming cube and applications, Geometry & Topology 20 (2016), 1427–1444.
Y. Benyamini and J. Lindenstrauss, Geometric Nonlinear Functional Analysis, American Mathematical Society Colloquium publications, Vol. 48, American Mathematical Society, Providence, RI, 2000.
Y. Benyamini, An extension theorem for separable Banach spaces, Israel Journal of Mathematics 29 (1978), 24–30.
C. Bessaga and A. Pelczyński, Spaces of continuous functions. IV. On isomorphical classification of spaces of continuous functions, Studia Mathematica 19 (1960), 53–62.
M. Cambern, On isomorphisms with small bound, Proceedings of the American Mathematical Society 18 (1967), 1062–1066.
H. B. Cohen, A bound-two isomorphism between C(X) Banach spaces, Proceedings of the American Mathematical Society 50 (1975), 215–217.
R. Deville, G. Godefroy and V. Zizler, Smoothness and Renormings in Banach Spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol. 64, Longman Scientific & Technical, Harlow, 1993.
Y. Dutrieux and N. Kalton, Perturbations of isometries between C(K)-spaces, Studia Mathematica 166 (2005), 181–197.
G. Godefroy, G. Lancien and V. Zizler, The non-linear geometry of Banach spaces after Nigel Kalton, Rocky Mountain Journal of Mathematics 44 (2014), 1529–1583.
R. Górak, Coarse version of the Banach–Stone theorem, Journal of Mathematical Analysis and Applications 377 (2011), 406–413.
P. Hájek, V. Montesinos-Santalucía, J. Vanderwerff and V. Zizler, Biorthogonal systems in Banach spaces, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, Vol. 26, Springer, New York, 2008.
K. Jarosz, Nonlinear generalizations of the Banach–Stone theorem, Studia Mathematica 93 (1989), 97–107.
N. Kalton and G. Lancien, Best constants for Lipschitz embeddings of metric spaces into c0, Fundamenta Mathematicae 199 (2008), 249–272.
G. Lancien, A survey on the Szlenk index and some of its applications, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas 100 (2006), 209–235.
C. J. Lennard, A. M. Tonge and A. Weston, Generalized roundness and negative type, Michigan Mathematical Journal 44 (1997), 37–45.
A. Pelczyński and Z. Semadeni, Spaces of continuous functions. III. Spaces C(O) for O without perfect subsets, Studia Mathematica 18 (1959), 211–222.
A. Procházka, Linear properties of Banach spaces and low distortion embeddings of metric graphs, arXiv: 1603.00741.
A. Procházka and L. Sánchez, Low distortion embeddings into Asplund Banach spaces, arXiv:1311.4584.
H. Rosenthal, The Banach Spaces C(K), in Handbook of the Geometry of Banach Spaces, Vol. 2, North-Holland, Amsterdam, 2003, pp. 1547–1602.
M. Zippin, The separable extension problem, Israel Journal of Mathematics 26 (1977), 372–387.
Author information
Authors and Affiliations
Corresponding author
Additional information
Partially supported by PHC Barrande 2013 26516YG and PEPS Insmi 2016.
Partially supported by MICINN Project MTM2012-34341 (Spain) and FONDE-CYT project 11130354 (Chile).
Rights and permissions
About this article
Cite this article
Procházka, A., Sánchez-González, L. Low distortion embeddings of some metric graphs into Banach spaces. Isr. J. Math. 220, 927–946 (2017). https://doi.org/10.1007/s11856-017-1525-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-017-1525-8