Banach May 2011

banach@mathdept.okstate.edu

1 participants
17 discussions

Abstract of a paper by A. Koldobsky, G. Paouris and M. Zymonopoulou
by alspach＠math.okstate.edu 23 May '11

23 May '11

This is an announcement for the paper "Isomorphic properties of intersection bodies" by A. Koldobsky, G. Paouris and M. Zymonopoulou. Abstract: We study isomorphic properties of two generalizations of intersection bodies, the class of k-intersection bodies and the class of generalized k-intersection bodies. We also show that the Banach-Mazur distance of the k-intersection body of a convex body, when it exists and it is convex, with the Euclidean ball, is bounded by a constant depending only on k, generalizing a well-known result of Hensley and Borell. We conclude by giving some volumetric estimates for k-intersection bodies. Archive classification: math.FA Submitted from: marisa.zym(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1105.2629 or http://arXiv.org/abs/1105.2629

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Abstract of a paper by Jose L. Gamez-Merino, Gustavo A. Munoz-Fernandez, Daniel Pellegrino and Juan B. Seoane-Sepulveda
by alspach＠math.okstate.edu 13 May '11

13 May '11

This is an announcement for the paper "Bounded and unbounded polynomials and multilinear forms: Characterizing continuity" by Jose L. Gamez-Merino, Gustavo A. Munoz-Fernandez, Daniel Pellegrino and Juan B. Seoane-Sepulveda. Abstract: In this paper we prove a characterization of continuity for polynomials on a normed space. Namely, we prove that a polynomial is continuous if and only if it maps compact sets into compact sets. We also provide a partial answer to the question as to whether a polynomial is continuous if and only if it transforms connected sets into connected sets. These results motivate the natural question as to how many non-continuous polynomials there are on an infinite dimensional normed space. A problem on the \emph{lineability} of the sets of non-continuous polynomials and multilinear mappings on infinite dimensional normed spaces is answered. Archive classification: math.FA Remarks: 8 pages Submitted from: dmpellegrino(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1105.1737 or http://arXiv.org/abs/1105.1737

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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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Banach May 2011