Banach

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Abstract of a paper by Valerio Capraro, Tobias Fritz
by alspach＠math.okstate.edu 13 May '11

13 May '11

This is an announcement for the paper "On the axiomatization of convex subsets of Banach spaces" by Valerio Capraro, Tobias Fritz. Abstract: We prove that any convex-like structure in the sense of Nate Brown is affinely and isometrically isomorphic to a closed convex subset of a Banach space. This answers an open question of Brown. As an intermediate step, we identify Brown's algebraic axioms as equivalent to certain well-known axioms of abstract convexity. Archive classification: math.MG math.FA math.OA Mathematics Subject Classification: Primary 52A01, Secondary 46L36 Remarks: 8 pages, 1 figure Submitted from: tobias.fritz(a)icfo.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1105.1270 or http://arXiv.org/abs/1105.1270

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Abstract of a paper by Elisabeth M. Werner
by alspach＠math.okstate.edu 13 May '11

13 May '11

This is an announcement for the paper "Renyi Divergence and $L_p$-affine surface area for convex bodies" by Elisabeth M. Werner. Abstract: We show that the fundamental objects of the $L_p$-Brunn-Minkowski theory, namely the $L_p$-affine surface areas for a convex body, are closely related to information theory: they are exponentials of R\'enyi divergences of the cone measures of a convex body and its polar. We give geometric interpretations for all R\'enyi divergences $D_\alpha$, not just for the previously treated special case of relative entropy which is the case $\alpha =1$. Now, no symmetry assumptions are needed and, if at all, only very weak regularity assumptions are required. Previously, the relative entropies appeared only after performing second order expansions of certain expressions. Now already first order expansions makes them appear. Thus, in the new approach we detect ``faster" details about the boundary of a convex body. Archive classification: math.FA Mathematics Subject Classification: 52A20, 53A15 Submitted from: elisabeth.werner(a)case.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1105.1124 or http://arXiv.org/abs/1105.1124

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Abstract of a paper by Rafal Gorak
by alspach＠math.okstate.edu 13 May '11

13 May '11

This is an announcement for the paper "Perturbations of isometries between Banach spaces" by Rafal Gorak. Abstract: We prove a very general theorem concerning the estimation of the expression \mbox{$\|T(\frac{a+b}{2}) - \frac{Ta+Tb}{2}\|$} for different kinds of maps $T$ satisfying some general perurbated isometry condition. It can be seen as a quantitative generalization of the classical Mazur-Ulam theorem. The estimates improve the existing ones for bi-Lipschitz maps. As a consequence we also obtain a very simple proof of the result of Gevirtz which answers the Hyers-Ulam problem and we prove a non-linear generalization of the Banach-Stone theorem which improves the results of Jarosz and more recent results of Dutrieux and Kalton. Archive classification: math.FA Mathematics Subject Classification: 46E40, 46B20 Submitted from: R.Gorak(a)mini.pw.edu.pl The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1105.0854 or http://arXiv.org/abs/1105.0854

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Abstract of a paper by Majid Gazor
by alspach＠math.okstate.edu 13 May '11

13 May '11

This is an announcement for the paper "Condensation rank of injective Banach spaces" by Majid Gazor. Abstract: The condensation rank associates any topological space with a unique ordinal number. In this paper we prove that the condensation rank of any infinite dimensional injective Banach space is equal to or greater than the first uncountable ordinal number. Archive classification: math.FA Mathematics Subject Classification: 46B25, 03E10, 54A05, 28A05 Submitted from: m.gazor.iut(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1104.4896 or http://arXiv.org/abs/1104.4896

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Abstract of a paper by Valentin Ferenczi and Christian Rosendal
by alspach＠math.okstate.edu 13 May '11

13 May '11

This is an announcement for the paper "Banach spaces without minimal subspaces - examples" by Valentin Ferenczi and Christian Rosendal. Abstract: We analyse several examples of separable Banach spaces, some of them new, and relate them to several dichotomies obtained in the previous paper spaces without minimal subspaces of the dichotomies they fall. This paper may be seen as a more empirical continuation of is on the study of examples for the new classes of Banach spaces considered in that work. Archive classification: math.FA Mathematics Subject Classification: 46B03, 03E15 Remarks: 29 pages, to appear in Annales de l Submitted from: ferenczi(a)ccr.jussieu.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1104.4724 or http://arXiv.org/abs/1104.4724

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Abstract of a paper by Almut Burchard and Marc Fortier
by alspach＠math.okstate.edu 29 Apr '11

29 Apr '11

This is an announcement for the paper "Convergence of random polarziations" by Almut Burchard and Marc Fortier. Abstract: We derive conditions under which random sequences of polarizations converge almost surely to the symmetric decreasing rearrangement. The parameters for the polarizations are independent random variables whose distributions may be far from uniform. The proof of convergence hinges on an estimate for the expected distance from the limit that also yields a bound on the rate of convergence. In the special case of i.i.d. sequences, we obtain almost sure convergence even for polarizations chosen at random from small sets. The precise characterization of convergent sequences remains an open problem. These statements about polarization allow us to improve the existing convergence results for Steiner symmetrization. In particular, we show that full rotational symmetry can be achieved by alternating Steiner symmetrization along directions that satisfy an explicit non-degeneracy condition. Finally, we construct examples for dense sequences of directions such that the corresponding Steiner symmetrizations do not converge. Archive classification: math.FA math.PR Mathematics Subject Classification: 60D05 (26D15, 28A75, 52A52) Remarks: 30 pages, 6 figures Submitted from: almut(a)math.toronto.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1104.4103 or http://arXiv.org/abs/1104.4103

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Abstract of a paper by Christian Rosendal
by alspach＠math.okstate.edu 29 Apr '11

29 Apr '11

This is an announcement for the paper "$\alpha$-minimal Banach spaces" by Christian Rosendal. Abstract: A Banach space with a Schauder basis is said to be $\alpha$-minimal for some countable ordinal $\alpha$ if, for any two block subspaces, the Bourgain embeddability index of one into the other is at least $\alpha$. We prove a dichotomy that characterises when a Banach space has an $\alpha$-minimal subspace, which contributes to the ongoing project, initiated by W. T. Gowers, of classifying separable Banach spaces by identifying characteristic subspaces. Archive classification: math.FA math.LO Mathematics Subject Classification: 46B03, 03E15 Submitted from: rosendal(a)math.uic.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1104.3543 or http://arXiv.org/abs/1104.3543

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Abstract of a paper by Hulya Duru, Arkady Kitover, and Mehmet Orhon
by alspach＠math.okstate.edu 29 Apr '11

29 Apr '11

This is an announcement for the paper "Multiplication operators on vector-valued function spaces" by Hulya Duru, Arkady Kitover, and Mehmet Orhon. Abstract: Let $E$ be a Banach function space on a probability measure space $(\Omega ,\Sigma,\mu).$ Let $X$ be a Banach space and $E(X)$ be the associated K\"{o}the-Bochner space. An operator on $E(X)$ is called a multiplication operator if it is given by multiplication by a function in $L^{\infty}(\mu).$ In the main result of this paper, we show that an operator $T$ on $E(X)$ is a multiplication operator if and only if $T$ commutes with $L^{\infty}(\mu)$ and leaves invariant the cyclic subspaces generated by the constant vector-valued functions in $E(X).$ As a corollary we show that this is equivalent to $T$ satisfying a functional equation considered by Calabuig, Rodr\'{i}guez, S\'{a}nchez-P\'{e}rez in [3]. Archive classification: math.FA Mathematics Subject Classification: 47B38 (Primary) 46G10, 46B42, 46H25 (Secondary) Submitted from: mo(a)unh.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1104.2806 or http://arXiv.org/abs/1104.2806

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Abstract of a paper by S. Lajara, A. Pallares and S. Troyanski
by alspach＠math.okstate.edu 29 Apr '11

29 Apr '11

This is an announcement for the paper "Moduli of convexity and smoothness of reflexive subspaces of L^1" by S. Lajara, A. Pallares and S. Troyanski. Abstract: We show that for any probability measure \mu there exists an equivalent norm on the space L^1(\mu) whose restriction to each reflexive subspace is uniformly smooth and uniformly convex, with modulus of convexity of power type 2. This renorming provides also an estimate for the corresponding modulus of smoothness of such subspaces. Archive classification: math.FA Mathematics Subject Classification: 46B03, 46B10, 46B20, 46B25 Submitted from: apall(a)um.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1104.2802 or http://arXiv.org/abs/1104.2802

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Abstract of a paper by Peter Elbau
by alspach＠math.okstate.edu 29 Apr '11

29 Apr '11

This is an announcement for the paper "Sequential lower semi-continuity of non-local functionals" by Peter Elbau. Abstract: We give a necessary and sufficient condition for non-local functionals on vector-valued Lebesgue spaces to be weakly sequentially lower semi-continuous. Here a non-local functional shall have the form of a double integral of a density which depends on the function values at two different points. The characterisation we get is essentially that the density has to be convex in one variable if we integrate over the other one with an arbitrary test function in it. Moreover, we show that this condition is in the case of non-local functionals on real-valued Lebesgue spaces (up to some equivalence in the density) equivalent to the separate convexity of the density. Archive classification: math.FA Mathematics Subject Classification: 49J05, 49J45 Remarks: 23 pages Submitted from: elbau(a)math.ethz.ch The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1104.2686 or http://arXiv.org/abs/1104.2686

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