Banach

banach@mathdept.okstate.edu

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Abstract of a paper by Emanuel Milman
by Dale Alspach 28 Apr '06

28 Apr '06

This is an announcement for the paper "On Gaussian marginals of uniformly convex bodies" by Emanuel Milman. Abstract: We show that many uniformly convex bodies have Gaussian marginals in most directions in a strong sense, which takes into account the tails of the distributions. These include uniformly convex bodies with power type 2, and power type p>2 with some additional type condition. In particular, all unit-balls of subspaces of L_p for 1<p<\infty have Gaussian marginals in this strong sense. Using the weaker Kolmogorov metric, we can extend our results to arbitrary uniformly convex bodies with power type p, for 2<=p<4. These results are obtained by putting the bodies in (surprisingly) non-isotropic positions and by a new concentration of volume observation for uniformly convex bodies. Archive classification: Functional Analysis; Metric Geometry; Probability Remarks: 21 pages The source file(s), Gaussian-Marginals.bbl: 5089 bytes, Gaussian-Marginals.tex: 76495 bytes, is(are) stored in gzipped form as 0604595.tar.gz with size 24kb. The corresponding postcript file has gzipped size 93kb. Submitted from: emanuel.milman(a)weizmann.ac.il The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0604595 or http://arXiv.org/abs/math.FA/0604595 or by email in unzipped form by transmitting an empty message with subject line uget 0604595 or in gzipped form by using subject line get 0604595 to: math(a)arXiv.org.

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Abstract of a paper by Boaz Klartag and Emanuel Milman
by Dale Alspach 28 Apr '06

28 Apr '06

This is an announcement for the paper "On volume distribution in 2-convex bodies" by Boaz Klartag and Emanuel Milman. Abstract: We consider convex sets whose modulus of convexity is uniformly quadratic. First, we observe several interesting relations between different positions of such ``2-convex'' bodies; in particular, the isotropic position is a finite volume-ratio position for these bodies. Second, we prove that high dimensional 2-convex bodies posses one-dimensional marginals that are approximately Gaussian. Third, we improve for 1<p<=2 some bounds on the isotropic constant of quotients of subspaces of L_p and S_p^m, the Schatten Class space. Archive classification: Functional Analysis; Metric Geometry Remarks: 27 pages The source file(s), 2-Convex-Bodies.bbl: 7979 bytes, 2-Convex-Bodies.tex: 70706 bytes, is(are) stored in gzipped form as 0604594.tar.gz with size 24kb. The corresponding postcript file has gzipped size 104kb. Submitted from: emanuel.milman(a)weizmann.ac.il The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0604594 or http://arXiv.org/abs/math.FA/0604594 or by email in unzipped form by transmitting an empty message with subject line uget 0604594 or in gzipped form by using subject line get 0604594 to: math(a)arXiv.org.

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Abstract of a paper by Marius Junge and Javier Parcet
by Dale Alspach 25 Apr '06

25 Apr '06

This is an announcement for the paper "Rosenthal's theorem for subspaces of noncommutative Lp" by Marius Junge and Javier Parcet. Abstract: We show that a reflexive subspace of the predual of a von Neumann algebra embeds into a noncommutative Lp space for some p>1. This is a noncommutative version of Rosenthal's result for commutative Lp spaces. Similarly for 1 < q < 2, an infinite dimensional subspace X of a noncommutative Lq space either contains lq or embeds in Lp for some q < p < 2. The novelty in the noncommutative setting is a double sided change of density. Archive classification: Functional Analysis; Operator Algebras Remarks: 34 pages The source file(s), Rosenthal.tex: 103990 bytes, is(are) stored in gzipped form as 0604510.gz with size 30kb. The corresponding postcript file has gzipped size 144kb. Submitted from: jparcet(a)crm.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0604510 or http://arXiv.org/abs/math.FA/0604510 or by email in unzipped form by transmitting an empty message with subject line uget 0604510 or in gzipped form by using subject line get 0604510 to: math(a)arXiv.org.

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Abstract of a paper by M. Mirzavaziri and M. S. Moslehian
by Dale Alspach 24 Apr '06

24 Apr '06

This is an announcement for the paper "Orthogonal constant mappings in isosceles orthogonal spaces" by M. Mirzavaziri and M. S. Moslehian. Abstract: In this paper we introduce the notion of orthogonally constant mapping in an isosceles orthogonal space and establish stability of orthogonally constant mappings. As an application, we discuss the orthogonal stability of the Pexiderized quadratic equation $f(x+y)+g(x+y)=h(x)+k(y)$. Archive classification: Classical Analysis and ODEs; Functional Analysis Mathematics Subject Classification: 39B55; 39B82; 39B52 Remarks: 7 pages, to appear in Kragujevac Math. J The source file(s), OrtCons_final.tex: 15092 bytes, is(are) stored in gzipped form as 0604463.gz with size 5kb. The corresponding postcript file has gzipped size 40kb. Submitted from: moslehian(a)ferdowsi.um.ac.ir The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.CA/0604463 or http://arXiv.org/abs/math.CA/0604463 or by email in unzipped form by transmitting an empty message with subject line uget 0604463 or in gzipped form by using subject line get 0604463 to: math(a)arXiv.org.

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Abstract of a paper by Shiri Artstein, Vitali D. Milman and Yaron Ostrover
by Dale Alspach 21 Apr '06

21 Apr '06

This is an announcement for the paper "The M-ellipsoid, symplectic capacities and volume" by Shiri Artstein, Vitali D. Milman and Yaron Ostrover. Abstract: In this work we bring together tools and ideology from two different fields, Symplectic Geometry and Asymptotic Geometric Analysis, to arrive at some new results. Our main result is a dimension-independent bound for the symplectic capacity of a convex body by its volume radius. Archive classification: Symplectic Geometry; Functional Analysis Mathematics Subject Classification: 53D05; 53C15; 46B07; 52A20; 46B20 The source file(s), CapMil2006Apr19.tex: 34307 bytes, is(are) stored in gzipped form as 0604434.gz with size 12kb. The corresponding postcript file has gzipped size 61kb. Submitted from: artstein(a)math.princeton.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.SG/0604434 or http://arXiv.org/abs/math.SG/0604434 or by email in unzipped form by transmitting an empty message with subject line uget 0604434 or in gzipped form by using subject line get 0604434 to: math(a)arXiv.org.

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Abstract of a paper by Roman Vershynin
by Dale Alspach 17 Apr '06

17 Apr '06

This is an announcement for the paper "Beyond Hirsch Conjecture: walks on random polytopes and smoothed complexity of the simplex method" by Roman Vershynin. Abstract: The smoothed analysis of algorithms is concerned with the expected running time of an algorithm under slight random perturbations of arbitrary inputs. Spielman and Teng proved that the shadow-vertex simplex method had polynomial smoothed complexity. On a slight random perturbation of arbitrary linear program, the simplex method finds the solution after a walk on polytope(s) with expected length polynomial in the number of constraints n, the number of variables d and the inverse standard deviation of the perturbation 1/sigma. We show that the length of walk in the simplex method is actually polylogarithmic in the number of constraints n. Spielman-Teng's bound on the walk was O(n^{86} d^{55} sigma^{-30}), up to logarithmic factors. We improve this to O(min(d^5 log^2(n), d^9 log^4(d), d^3 sigma^{-4})). This shows that the tight Hirsch conjecture n-d on the the length of walk on polytopes is not a limitation for the smoothed Linear Programming. Random perturbations create short paths between vertices. We propose a randomized phase-I for solving arbitrary linear programs. Instead of finding a vertex of a feasible set, we add a vertex at random to the feasible set. This does not affect the solution of the linear program with constant probability. So, in expectation it takes a constant number of independent trials until a correct solution is found. This overcomes one of the major difficulties of smoothed analysis of the simplex method -- one can now statistically decouple the walk from the smoothed linear program. This yields a much better reduction of the smoothed complexity to a geometric quantity -- the size of planar sections of random polytopes. We also improve upon the known estimates for that size. Archive classification: Data Structures and Algorithms; Functional Analysis Remarks: 17 pages The source file(s), , is(are) stored in gzipped form as with size . The corresponding postcript file has gzipped size . Submitted from: vershynin(a)math.ucdavis.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/cs.DS/0604055 or http://arXiv.org/abs/cs.DS/0604055 or by email in unzipped form by transmitting an empty message with subject line uget 04055 or in gzipped form by using subject line get 04055 to: math(a)arXiv.org.

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Abstract of a paper by Apostolos Giannopoulos, Alain Pajor, and Grigoris Paouris
by Dale Alspach 17 Apr '06

17 Apr '06

This is an announcement for the paper "A note on subgaussian estimates for linear functionals on convex bodies" by Apostolos Giannopoulos, Alain Pajor, and Grigoris Paouris. Abstract: We give an alternative proof of a recent result of Klartag on the existence of almost subgaussian linear functionals on convex bodies. If $K$ is a convex body in ${\mathbb R}^n$ with volume one and center of mass at the origin, there exists $x\neq 0$ such that $$|\{ y\in K:\,|\langle y,x\rangle |\gr t\|\langle\cdot ,x\rangle\|_1\}|\ls\exp (-ct^2/\log^2(t+1))$$ for all $t\gr 1$, where $c>0$ is an absolute constant. The proof is based on the study of the $L_q$--centroid bodies of $K$. Analogous results hold true for general log-concave measures. Archive classification: Functional Analysis; Metric Geometry Mathematics Subject Classification: 46B07, 52A20 Remarks: 10 pages The source file(s), subgaussian.tex: 24859 bytes, is(are) stored in gzipped form as 0604299.gz with size 8kb. The corresponding postcript file has gzipped size 54kb. Submitted from: apgiannop(a)math.uoa.gr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0604299 or http://arXiv.org/abs/math.FA/0604299 or by email in unzipped form by transmitting an empty message with subject line uget 0604299 or in gzipped form by using subject line get 0604299 to: math(a)arXiv.org.

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Abstract of a paper by Vladimir Kadets, Miguel Martin, and Javier Meri
by Dale Alspach 06 Apr '06

06 Apr '06

This is an announcement for the paper "Norm equalities for operators" by Vladimir Kadets, Miguel Martin, and Javier Meri. Abstract: A Banach space $X$ has the Daugavet property if the Daugavet equation $\|\Id + T\|= 1 + \|T\|$ holds for every rank-one operator $T:X \longrightarrow X$. We show that the most natural attempts to introduce new properties by considering other norm equalities for operators (like $\|g(T)\|=f(\|T\|)$ for some functions $f$ and $g$) lead in fact to the Daugavet property of the space. On the other hand there are equations (for example $\|\Id + T\|= \|\Id - T\|$) that lead to new, strictly weaker properties of Banach spaces. Archive classification: Functional Analysis Mathematics Subject Classification: 46B20 Remarks: 21 pages The source file(s), KadMarMer.tex: 56515 bytes, is(are) stored in gzipped form as 0604102.gz with size 17kb. The corresponding postcript file has gzipped size 87kb. Submitted from: mmartins(a)ugr.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0604102 or http://arXiv.org/abs/math.FA/0604102 or by email in unzipped form by transmitting an empty message with subject line uget 0604102 or in gzipped form by using subject line get 0604102 to: math(a)arXiv.org.

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Abstract of a paper by Olivier Guedon and Mark Rudelson
by Dale Alspach 05 Apr '06

05 Apr '06

This is an announcement for the paper "L_p moments of random vectors via majorizing measures" by Olivier Guedon and Mark Rudelson. Abstract: For a random vector X in R^n, we obtain bounds on the size of a sample, for which the empirical p-th moments of linear functionals are close to the exact ones uniformly on an n-dimensional convex body K. We prove an estimate for a general random vector and apply it to several problems arising in geometric functional analysis. In particular, we find a short Lewis type decomposition for any finite dimensional subspace of L_p. We also prove that for an isotropic log-concave random vector, we only need about n^{p/2} \log n sample points so that the empirical p-th moments of the linear functionals are almost isometrically the same as the exact ones. We obtain a concentration estimate for the empirical moments. The main ingredient of the proof is the construction of an appropriate majorizing measure to bound a certain Gaussian process. Archive classification: Functional Analysis Mathematics Subject Classification: 46B09, 52A21 Remarks: 32 pages, to appear in Advances in Mathematics The source file(s), ADVgr06-03-15.tex: 71461 bytes, is(are) stored in gzipped form as 0507023.gz with size 21kb. The corresponding postcript file has gzipped size 108kb. Submitted from: rudelson(a)math.missouri.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0507023 or http://arXiv.org/abs/math.FA/0507023 or by email in unzipped form by transmitting an empty message with subject line uget 0507023 or in gzipped form by using subject line get 0507023 to: math(a)arXiv.org.

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Abstract of a paper by Konrad J Swanepoel and Rafael Villa
by Dale Alspach 28 Mar '06

28 Mar '06

This is an announcement for the paper "A lower bound for the equilateral number of normed spaces" by Konrad J Swanepoel and Rafael Villa. Abstract: We show that if the Banach-Mazur distance between an n-dimensional normed space X and ell infinity is at most 3/2, then there exist n+1 equidistant points in X. By a well-known result of Alon and Milman, this implies that an arbitrary n-dimensional normed space admits at least e^{c sqrt(log n)} equidistant points, where c>0 is an absolute constant. We also show that there exist n equidistant points in spaces sufficiently close to n-dimensional ell p (1 < p < infinity). Archive classification: Metric Geometry; Functional Analysis Mathematics Subject Classification: 46B04 (Primary); 46B20, 52A21, 52C17 (Secondary) Remarks: 5 pages The source file(s), equilateral-lower3.tex: 14633 bytes, is(are) stored in gzipped form as 0603614.gz with size 5kb. The corresponding postcript file has gzipped size 39kb. 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/0603614 or http://arXiv.org/abs/math.MG/0603614 or by email in unzipped form by transmitting an empty message with subject line uget 0603614 or in gzipped form by using subject line get 0603614 to: math(a)arXiv.org.

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