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
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to: math(a)arXiv.org.
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
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