Banach

banach@mathdept.okstate.edu

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Abstract of a paper by Matthieu Fradelizi and Mathieu Meyer
by Dale Alspach 06 Oct '06

06 Oct '06

This is an announcement for the paper "Some functional forms of Blaschke-Santal\'o inequality" by Matthieu Fradelizi and Mathieu Meyer. Abstract: We establish new functional versions of the Blaschke-Santal\'o inequality on the volume product of a convex body which generalize to the non-symmetric setting an inequality of K.~Ball and we give a simple proof of the case of equality. As a corollary, we get some inequalities for $\log$-concave functions and Legendre transforms which extend the recent result of Artstein, Klartag and Milman, with its equality case. Archive classification: Functional Analysis; Metric Geometry Mathematics Subject Classification: 52A40 Remarks: 19 pages, to appear in Mathematische Zeitschrift The source file(s), Blaschke-Santalo-final.tex: 48038 bytes, is(are) stored in gzipped form as 0609553.gz with size 15kb. The corresponding postcript file has gzipped size 71kb. Submitted from: matthieu.fradelizi(a)univ-mlv.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0609553 or http://arXiv.org/abs/math.FA/0609553 or by email in unzipped form by transmitting an empty message with subject line uget 0609553 or in gzipped form by using subject line get 0609553 to: math(a)arXiv.org.

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Abstract of a paper by Konrad J. Swanepoel
by Dale Alspach 06 Oct '06

06 Oct '06

This is an announcement for the paper "A problem of Kusner on equilateral sets" by Konrad J. Swanepoel. Abstract: R. B. Kusner [R. Guy, Amer. Math. Monthly 90 (1983), 196--199] asked whether a set of vectors in a d-dimensional real vector space such that the l-p distance between any pair is 1, has cardinality at most d+1. We show that this is true for p=4 and any d >= 1, and false for all 1<p<2 with d sufficiently large, depending on p. More generally we show that the maximum cardinality is at most $(2\lceil p/4\rceil-1)d+1$ if p is an even integer, and at least $(1+\epsilon_p)d$ if 1<p<2, where $\epsilon_p>0$ depends on p. Archive classification: Metric Geometry; Functional Analysis Mathematics Subject Classification: 52C10 (Primary) 52A21, 46B20 (Secondary) Citation: Archiv der Mathematik (Basel) 83 (2004), no. 2, 164--170 Remarks: 6 pages. Small correction to Proposition 2 The source file(s), kusner-corrected.tex: 19322 bytes, is(are) stored in gzipped form as 0309317.gz with size 7kb. The corresponding postcript file has gzipped size 43kb. Submitted from: swanekj(a)unisa.ac.za The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.MG/0309317 or http://arXiv.org/abs/math.MG/0309317 or by email in unzipped form by transmitting an empty message with subject line uget 0309317 or in gzipped form by using subject line get 0309317 to: math(a)arXiv.org.

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Abstract of a paper by Konrad J. Swanepoel
by Dale Alspach 27 Sep '06

27 Sep '06

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Abstract of a paper by Konrad J. Swanepoel
by Dale Alspach 22 Sep '06

22 Sep '06

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Abstract of a paper by George Androulakis, Gleb Sirotkin, and Vladimir G. Troitsky
by Dale Alspach 05 Sep '06

05 Sep '06

This is an announcement for the paper "Classes of strictly singular operators and their products" by George Androulakis, Gleb Sirotkin, and Vladimir G. Troitsky. Abstract: V.~D. Milman proved in~\cite{Milman:70} that the product of two strictly singular operators on $L_p[0,1]$ ($1\le p<\infty$) or on $C[0,1]$ is compact. In this note we utilize Schreier families $\mathcal{S}_\xi$ in order to define the class of $\mathcal{S}_\xi $-strictly singular operators, and then we refine the technique of Milman to show that certain products of operators from this class are compact, under the assumption that the underlying Banach space has finitely many equivalence classes of Schreier-spreading sequences. Finally we define the class of ${\mathcal S}_\xi$-hereditarily indecomposable Banach spaces and we examine the operators on them. Archive classification: Functional Analysis Mathematics Subject Classification: 47B07, 47A15 The source file(s), compactproducts.tex: 76155 bytes, is(are) stored in gzipped form as 0609039.gz with size 22kb. The corresponding postcript file has gzipped size 102kb. Submitted from: giorgis(a)math.sc.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0609039 or http://arXiv.org/abs/math.FA/0609039 or by email in unzipped form by transmitting an empty message with subject line uget 0609039 or in gzipped form by using subject line get 0609039 to: math(a)arXiv.org.

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Abstract of a paper by M.Cencelj, J.Dydak, J.Smrekar, and A.Vavpetic
by Dale Alspach 29 Aug '06

29 Aug '06

This is an announcement for the paper "Sublinear Higson corona and Lipschitz extensions" by M.Cencelj, J.Dydak, J.Smrekar, and A.Vavpetic. Abstract: The purpose of the paper is to characterize the dimension of sublinear Higson corona $\nu_L(X)$ of $X$ in terms of Lipschitz extensions of functions: Theorem: Suppose $(X,d)$ is a proper metric space. The dimension of the sublinear Higson corona $\nu_L(X)$ of $X$ is the smallest integer $m\ge 0$ with the following property: Any norm-preserving asymptotically Lipschitz function $f'\colon A\to \R^{m+1}$, $A\subset X$, extends to a norm-preserving asymptotically Lipschitz function $g'\colon X\to \R^{m+1}$. One should compare it to the result of Dranishnikov \cite{Dr1} who characterized the dimension of the Higson corona $\nu(X)$ of $X$ is the smallest integer $n\ge 0$ such that $\R^{n+1}$ is an absolute extensor of $X$ in the asymptotic category $\AAA$ (that means any proper asymptotically Lipschitz function $f\colon A\to \R^{n+1}$, $A$ closed in $X$, extends to a proper asymptotically Lipschitz function $f'\colon X\to \R^{n+1}$). \par In \cite{Dr1} Dranishnikov introduced the category $\tilde \AAA$ whose objects are pointed proper metric spaces $X$ and morphisms are asymptotically Lipschitz functions $f\colon X\to Y$ such that there are constants $b,c > 0$ satisfying $|f(x)|\ge c\cdot |x|-b$ for all $x\in X$. We show $\dim(\nu_L(X))\leq n$ if and only if $\R^{n+1}$ is an absolute extensor of $X$ in the category $\tilde\AAA$. \par As an application we reprove the following result of Dranishnikov and Smith \cite{DRS}: Theorem: Suppose $(X,d)$ is a proper metric space of finite asymptotic Assouad-Nagata dimension $\asdim_{AN}(X)$. If $X$ is cocompact and connected, then $\asdim_{AN}(X)$ equals the dimension of the sublinear Higson corona $\nu_L(X)$ of $X$. Archive classification: Metric Geometry; Functional Analysis; Geometric Topology Remarks: 13 pages The source file(s), SublinearHigson.tex: 51559 bytes, is(are) stored in gzipped form as 0608686.gz with size 15kb. The corresponding postcript file has gzipped size 76kb. Submitted from: dydak(a)math.utk.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.MG/0608686 or http://arXiv.org/abs/math.MG/0608686 or by email in unzipped form by transmitting an empty message with subject line uget 0608686 or in gzipped form by using subject line get 0608686 to: math(a)arXiv.org.

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Abstract of a paper by Shahar Mendelson, Alain Pajor and Nicole Tomczak-Jaegermann
by Dale Alspach 29 Aug '06

29 Aug '06

This is an announcement for the paper "Uniform uncertainty principle for Bernoulli and subgaussian ensembles" by Shahar Mendelson, Alain Pajor and Nicole Tomczak-Jaegermann. Abstract: We present a simple solution to a question posed by Candes, Romberg and Tao on the uniform uncertainty principle for Bernoulli random matrices. More precisely, we show that a rectangular k*n random subgaussian matrix (with k < n) has the property that by arbitrarily extracting any m (with m < k) columns, the resulting submatrices are arbitrarily close to (multiples of) isometries of a Euclidean space. We obtain the optimal estimate for m as a function of k,n and the degree of "closeness" to an isometry. We also give a short and self-contained solution of the reconstruction problem for sparse vectors. Archive classification: Statistics; Functional Analysis Mathematics Subject Classification: 46B07; 47B06; 41A05; 62G05; 94B75 Remarks: 15 pages; no figures; submitted The source file(s), uup-arx-21-08.tex: 48079 bytes, is(are) stored in gzipped form as 0608665.gz with size 16kb. The corresponding postcript file has gzipped size 71kb. Submitted from: alain.pajor(a)univ-mlv.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.ST/0608665 or http://arXiv.org/abs/math.ST/0608665 or by email in unzipped form by transmitting an empty message with subject line uget 0608665 or in gzipped form by using subject line get 0608665 to: math(a)arXiv.org.

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Abstract of a paper by Christian Rosendal
by Dale Alspach 25 Aug '06

25 Aug '06

This is an announcement for the paper "Infinite asymptotic games" by Christian Rosendal. Abstract: We study infinite asymptotic games in Banach spaces with an F.D.D. and prove that analytic games are determined by characterising precisely the conditions for the players to have winning strategies. These results are applied to characterise spaces embeddable into $\ell_p$ sums of finite dimensional spaces, extending results of Odell and Schlumprecht, and to study various notions of homogeneity of bases and Banach spaces. These results are related to questions of rapidity of subsequence extraction from normalised weakly null sequences. Archive classification: Functional Analysis; Logic Mathematics Subject Classification: Primary: 46B03, Secondary 03E15 The source file(s), AsymptoticGames18.tex: 61261 bytes, is(are) stored in gzipped form as 0608616.gz with size 19kb. The corresponding postcript file has gzipped size 83kb. Submitted from: rosendal(a)math.uiuc.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0608616 or http://arXiv.org/abs/math.FA/0608616 or by email in unzipped form by transmitting an empty message with subject line uget 0608616 or in gzipped form by using subject line get 0608616 to: math(a)arXiv.org.

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Abstract of a paper by S. J. Dilworth, V. Ferenczi, Denka Kutzarova and E. Odell
by Dale Alspach 23 Aug '06

23 Aug '06

This is an announcement for the paper "On strongly asymptotic $\ell_p$ spaces and minimality" by S. J. Dilworth, V. Ferenczi, Denka Kutzarova and E. Odell. Abstract: We study Banach spaces X with a strongly asymptotic l_p basis (any disjointly supported finite set of vectors far enough out with respect to the basis behaves like l_p) which are minimal (X embeds into every infinite dimensional subspace). In particular such spaces embed into l_p. Archive classification: Functional Analysis Mathematics Subject Classification: 46B20, 46B45 Remarks: 12 pages, AMSLaTeX The source file(s), dfko010206-archive.tex: 46987 bytes, is(are) stored in gzipped form as 0608550.gz with size 15kb. The corresponding postcript file has gzipped size 71kb. Submitted from: combs(a)mail.ma.utexas.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0608550 or http://arXiv.org/abs/math.FA/0608550 or by email in unzipped form by transmitting an empty message with subject line uget 0608550 or in gzipped form by using subject line get 0608550 to: math(a)arXiv.org.

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Abstract of a paper by Mohsen Alimohammady
by Dale Alspach 02 Aug '06

02 Aug '06

This is an announcement for the paper "Containment of $\c_{\bf 0}$ and $\ell_{\bf 1}$ in $\Pi_{\bf 1} \hbox{\bf (}\E\hbox{\bf ,}\ \F\hbox{\bf )}$" by Mohsen Alimohammady. Abstract: Suppose $\Pi_{1} (E, F)$ is the space of all absolutely 1-summing operators between two Banach spaces $E$ and $F$. We show that if $F$ has a copy of $c_{0}$, then $\Pi_{1} (E, F)$ will have a copy of $c_{0}$, and under some conditions if $E$ has a copy of $\ell_{1}$ then $\Pi_{1} (E, F)$ would have a complemented copy of $\ell_{1}$. Archive classification: Functional Analysis Mathematics Subject Classification: 47B10; 46B20 Remarks: 4 pages The source file(s), mat01.cls: 37258 bytes, mathtimy.sty: 20 bytes, pm2197new.tex: 11816 bytes, is(are) stored in gzipped form as 0607651.tar.gz with size 14kb. The corresponding postcript file has gzipped size 25kb. Submitted from: amohsen(a)umz.ac.ir The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0607651 or http://arXiv.org/abs/math.FA/0607651 or by email in unzipped form by transmitting an empty message with subject line uget 0607651 or in gzipped form by using subject line get 0607651 to: math(a)arXiv.org.

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