Diamond graphs and super-reflexivity. (English) Zbl 1183.46022
The main result of the paper is: A Banach space \(X\) is not super-reflexive if and only if the diamond graphs \(D_n\) Lipschitz embed into \(X\) with distortions independent of \(n\), and a similar result for Laakso graphs. Diamond graphs are defined in the following way: the first diamond graph \(D_0\) is a graph with two vertices and one edge between them. Each next diamond graph is obtained from the previous one if we replace each of the edges in the previous graph by a \(4\)-cycle with two of its opposite vertices being the end vertices of the edge and the two remaining vertices being new vertices. It seems that the diamond graphs were for the first time introduced into metric geometry in the paper of [A.Gupta, I.Newman, Y.Rabinovich and A.Sinclair, “Cuts, trees and \(\ell_1\)-embeddings of graphs”, Combinatorica 24, No.2, 233–269 (2004; Zbl 1056.05040)]. Laakso graphs are defined similarly, in a slightly more complicated way, they were introduced in [T.J.Laakso, Geom.Funct.Anal.10, No.1, 111–123 (2000); erratum ibid.12, 650 (2002; Zbl 0962.30006)]. Laakso graphs are of interest in the present context because they have uniformly bounded geometry and are uniformly doubling.
A similar characterization in terms of dyadic binary trees was obtained by J.Bourgain [Isr.J.Math.56, 222–230 (1986; Zbl 0643.46013)]; an “infinite” version of Bourgain’s characterization was found by F.Baudier [Arch.Math.89, No.5, 419–429 (2007; Zbl 1142.46007)].
One of the consequences of the result on diamond graphs and previously known results is that dimension reduction in the spirit of the paper [W.B.Johnson and J.Lindenstrauss, Contemp.Math.26, 189–206 (1984; Zbl 0539.46017)] fails in any non-super-reflexive space with non trivial type.
The authors also introduce the concept of Lipschitz \((p,r)\)-summing map and prove that every Lipschitz mapping is Lipschitz \((p,r)\)-summing for every \(1\leq r < p\). This result contrasts with a recent result of J.Farmer and W.B.Johnson [Proc.Am.Math.Soc.137, No.9, 2989–2995 (2009; Zbl 1183.46020)] where is was proved that a bounded linear operator is Lipschitz \(p\)-summing if and only if it is \(p\)-summing.
A similar characterization in terms of dyadic binary trees was obtained by J.Bourgain [Isr.J.Math.56, 222–230 (1986; Zbl 0643.46013)]; an “infinite” version of Bourgain’s characterization was found by F.Baudier [Arch.Math.89, No.5, 419–429 (2007; Zbl 1142.46007)].
One of the consequences of the result on diamond graphs and previously known results is that dimension reduction in the spirit of the paper [W.B.Johnson and J.Lindenstrauss, Contemp.Math.26, 189–206 (1984; Zbl 0539.46017)] fails in any non-super-reflexive space with non trivial type.
The authors also introduce the concept of Lipschitz \((p,r)\)-summing map and prove that every Lipschitz mapping is Lipschitz \((p,r)\)-summing for every \(1\leq r < p\). This result contrasts with a recent result of J.Farmer and W.B.Johnson [Proc.Am.Math.Soc.137, No.9, 2989–2995 (2009; Zbl 1183.46020)] where is was proved that a bounded linear operator is Lipschitz \(p\)-summing if and only if it is \(p\)-summing.
Reviewer: Mikhail Ostrovskii (Queens)
MSC:
| 46B85 | Embeddings of discrete metric spaces into Banach spaces; applications in topology and computer science |
| 05C12 | Distance in graphs |
| 46B07 | Local theory of Banach spaces |
| 46B10 | Duality and reflexivity in normed linear and Banach spaces |
Keywords:
absolutely summing operator; diamond graph; dimension reduction; Lipschitz embedding; super-reflexive Banach spaceCitations:
Zbl 0962.30006; Zbl 0643.46013; Zbl 1142.46007; Zbl 0539.46017; Zbl 1183.46020; Zbl 1056.05040References:
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