Geodesic
The metric geometry of the Hamming cube and applications
Baudier, Florent1 ; Freeman, Daniel2 ; Schlumprecht, Thomas3 ; Zsák, András4
1 Department of Mathematics, Texas A&M University, College Station, TX 77843, USA, Institut de Mathématiques Jussieu-Paris Rive Gauche, Université Pierre et Marie Curie, 75005 Paris, France
2 Department of Mathematics and Computer Science, Saint Louis University, 220 N. Grand Blvd., St. Louis, MO 63103, USA
3 Department of Mathematics, Texas A&M University, College Station, TX 77843, USA, Faculty of Electrical Engineering, Czech Technical University in Prague, Zikova 4, 166 27 Prague, Czech Republic
4 Peterhouse, University of Cambridge, Cambridge CB2 1RD, UK
Geometry & topology, Tome 20 (2016) no. 3, pp. 1427-1444.

Voir la notice de l'article provenant de la source Mathematical Sciences Publishers

The Lipschitz geometry of segments of the infinite Hamming cube is studied. Tight estimates on the distortion necessary to embed the segments into spaces of continuous functions on countable compact metric spaces are given. As an application, the first nontrivial lower bounds on the C(K)–distortion of important classes of separable Banach spaces, where K is a countable compact space in the family {[0,ω],[0,ω ⋅ 2],…,[0,ω2],…,[0,ωk ⋅ n],…,[0,ωω]} are obtained.

DOI : 10.2140/gt.2016.20.1427
Classification : 46B20, 46B80, 46B85
Keywords: countable compact metric space, bi-Lipschitz embedding, $C(K)$ space

Baudier, Florent 1 ; Freeman, Daniel 2 ; Schlumprecht, Thomas 3 ; Zsák, András 4

1 Department of Mathematics, Texas A&M University, College Station, TX 77843, USA, Institut de Mathématiques Jussieu-Paris Rive Gauche, Université Pierre et Marie Curie, 75005 Paris, France
2 Department of Mathematics and Computer Science, Saint Louis University, 220 N. Grand Blvd., St. Louis, MO 63103, USA
3 Department of Mathematics, Texas A&M University, College Station, TX 77843, USA, Faculty of Electrical Engineering, Czech Technical University in Prague, Zikova 4, 166 27 Prague, Czech Republic
4 Peterhouse, University of Cambridge, Cambridge CB2 1RD, UK 
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Baudier, Florent; Freeman, Daniel; Schlumprecht, Thomas; Zsák, András. The metric geometry of the Hamming cube and applications. Geometry & topology, Tome 20 (2016) no. 3, pp. 1427-1444. doi : 10.2140/gt.2016.20.1427. http://geodesic.mathdoc.fr/articles/10.2140/gt.2016.20.1427/

[1] I Aharoni, Every separable metric space is Lipschitz equivalent to a subset of c0+, Israel J. Math. 19 (1974) 284

[2] F Albiac, N J Kalton, Topics in Banach space theory, 233, Springer (2006)

[3] P Assouad, Remarques sur un article de Israel Aharoni sur les prolongements lipschitziens dans c0 (Israel J. Math. 19 (1974) 284–291), Israel J. Math. 31 (1978) 97

[4] S Banach, S Mazur, Zur Theorie der linearen Dimension, Stud. Math. 4 (1933) 100

[5] J Bourgain, The Szlenk index and operators on C(K)–spaces, Bull. Soc. Math. Belg. Sér. B 31 (1979) 87

[6] M M Fréchet, Sur quelques points du calcul fonctionnel, Rend. Circ. Mat Palermo Math. 22 (1906) 1

[7] G Godefroy, N J Kalton, Lipschitz-free Banach spaces, Studia Math. 159 (2003) 121

[8] N J Kalton, G Lancien, Best constants for Lipschitz embeddings of metric spaces into c0, Fund. Math. 199 (2008) 249

[9] E Odell, Ordinal indices in Banach spaces, Extracta Math. 19 (2004) 93

[10] J Pelant, Embeddings into c0, Topology Appl. 57 (1994) 259

[11] A Procházka, L Sánchez-González, Low distortion embeddings into Asplund Banach spaces, preprint (2014)

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