Banach August 2004

banach@mathdept.okstate.edu

1 participants
2 discussions

Abstract of a paper by Steven F. Bellenot
by Dale Alspach 03 Aug '04

03 Aug '04

This is an announcement for the paper "Skipped blocking and other decompositions in Banach spaces" by Steven F. Bellenot. Abstract: Necessary and sufficient conditions are given for when a sequence of finite dimensional subspaces (X_n) can be blocked to be a skipped blocking decompositon (SBD). The condition is order independent, so permutations of conditional basis, for example can be so blocked. This condition is implied if (X_n) is shrinking, or (X_n) is a permutation of a FDD, or if X is reflexive and (X_n) is separating. A separable space X has PCP, if and only if, every norming decomposition (X_n) can be blocked to be a boundedly complete SBD. Every boundedly complete SBD is a JT-decomposition. Archive classification: Functional Analysis Mathematics Subject Classification: 46B20 (Primary); 46B15, 46B22 (Secondary) Report Number: FSU04-11 Remarks: 11 pages, 0 figures The source file(s), skipB.tex: 42550 bytes, is(are) stored in gzipped form as 0408004.gz with size 13kb. The corresponding postcript file has gzipped size 65kb. Submitted from: bellenot(a)math.fsu.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0408004 or http://arXiv.org/abs/math.FA/0408004 or by email in unzipped form by transmitting an empty message with subject line uget 0408004 or in gzipped form by using subject line get 0408004 to: math(a)arXiv.org.

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Abstract of a paper by J. R. Lee and A. Naor
by Dale Alspach 02 Aug '04

02 Aug '04

This is an announcement for the paper "Embedding the diamond graph in $L_p$ and dimension reduction in $L_1$" by J. R. Lee and A. Naor. Abstract: We show that any embedding of the level-k diamond graph of Newman and Rabinovich into $L_p$, $1 < p \le 2$, requires distortion at least $\sqrt{k(p-1) + 1}$. An immediate consequence is that there exist arbitrarily large n-point sets $X \subseteq L_1$ such that any D-embedding of X into $\ell_1^d$ requires $d \geq n^{\Omega(1/D^2)}$. This gives a simple proof of the recent result of Brinkman and Charikar which settles the long standing question of whether there is an $L_1$ analogue of the Johnson-Lindenstrauss dimension reduction lemma. Archive classification: Functional Analysis; Combinatorics; Metric Geometry Remarks: 3 pages. To appear in Geometric and Functional Analysis (GAFA) The source file(s), diamond-gafa.tex: 8222 bytes, is(are) stored in gzipped form as 0407520.gz with size 3kb. The corresponding postcript file has gzipped size 31kb. Submitted from: jrl(a)cs.berkeley.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0407520 or http://arXiv.org/abs/math.FA/0407520 or by email in unzipped form by transmitting an empty message with subject line uget 0407520 or in gzipped form by using subject line get 0407520 to: math(a)arXiv.org.

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Banach August 2004