Banach

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Abstract of a paper by Matthieu Fradelizi and Arnaud Marsiglietti
by alspach＠math.okstate.edu 27 Feb '13

27 Feb '13

This is an announcement for the paper "On the analogue of the concavity of entropy power in the Brunn-Minkowski theory" by Matthieu Fradelizi and Arnaud Marsiglietti. Abstract: Elaborating on the similarity between the entropy power inequality and the Brunn-Minkowski inequality, Costa and Cover conjectured in {\it On the similarity of the entropy power inequality and the Brunn-Minkowski inequality} (IEEE Trans. Inform. Theory 30 (1984), no. 6, 837-839) the $\frac{1}{n}$-concavity of the outer parallel volume of measurable sets as an analogue of the concavity of entropy power. We investigate this conjecture and study its relationship with geometric inequalities. Archive classification: math.FA cs.IT math.IT math.MG Mathematics Subject Classification: 52A40, 94A17 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/1302.6093 or http://arXiv.org/abs/1302.6093

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Abstract of a paper by Oleg Reinov and Asfand Fahad
by alspach＠math.okstate.edu 27 Feb '13

27 Feb '13

This is an announcement for the paper "On dentability in locally convex vector spaces" by Oleg Reinov and Asfand Fahad. Abstract: For a locally convex vector space (l.c.v.s.) $E$ and an absolutely convex neighborhood $V$ of zero, a bounded subset $A$ of $E$ is said to be $V$-dentable (respectively, $V$-f-dentable) if for any $\epsilon>0$ there exists an $x\in A$ so that $$x\notin \overline{co} (A\setminus (x+\epsilon V)) $$ (respectively, so that $$ x\notin {co} (A\setminus (x+\epsilon V)) ). $$ Here, "$\overline{co}$" denotes the closure in $E$ of the convex hull of a set. We present a theorem which says that for a wide class of bounded subsets $B$ of locally convex vector spaces the following is true: $(V)$ every subset of $B$ is $V$-dentable if and only if every subset of $B$ is $V$-f-dentable. The proof is purely geometrical and independent of any related facts. As a consequence (in the particular case where $B$ is complete convex bounded metrizable subset of a l.c.v.s.), we obtain a positive solution to a 1978-hypothesis of Elias Saab (see p. 290 in "On the Radon-Nikodym property in a class of locally convex spaces", Pacific J. Math. 75, No. 1, 1978, 281-291). Archive classification: math.FA Remarks: 5 pages, AMSTeX Submitted from: orein51(a)mail.ru The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1302.6019 or http://arXiv.org/abs/1302.6019

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Abstract of a paper by Mikhail I. Ostrovskii
by alspach＠math.okstate.edu 27 Feb '13

27 Feb '13

This is an announcement for the paper "On metric characterizations of the Radon-Nikodym and related properties of Banach spaces" by Mikhail I. Ostrovskii. Abstract: We find a class of metric structures which do not admit bilipschitz embeddings into Banach spaces with the Radon-Nikod\'ym property. Our proof relies on Chatterji's (1968) martingale characterization of the RNP and does not use the Cheeger's (1999) metric differentiation theory. The class includes the infinite diamond and both Laakso (2000) spaces. We also show that for each of these structures there is a non-RNP Banach space which does not admit its bilipschitz embedding. We prove that a dual Banach space does not have the RNP if and only if it admits a bilipschitz embedding of the infinite diamond. The paper also contains related characterizations of reflexivity and the infinite tree property. Archive classification: math.FA math.MG Mathematics Subject Classification: Primary: 46B22, Secondary: 05C12, 30L05, 46B10, 46B85, 54E35 Submitted from: ostrovsm(a)stjohns.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1302.5968 or http://arXiv.org/abs/1302.5968

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Abstract of a paper by Jaegil Kim and Artem Zvavitch
by alspach＠math.okstate.edu 27 Feb '13

27 Feb '13

This is an announcement for the paper "Stability of the reverse Blaschke-Santalo inequality for unconditional convex bodies" by Jaegil Kim and Artem Zvavitch. Abstract: Mahler's conjecture asks whether the cube is a minimizer for the volume product of a body and its polar in the class of symmetric convex bodies in R^n. The corresponding inequality to the conjecture is sometimes called the the reverse Blaschke-Santalo inequality. The conjecture is known in dimension two and in several special cases. In the class of unconditional convex bodies, Saint Raymond confirmed the conjecture, and Meyer and Reisner, independently, characterized the equality case. In this paper we present a stability version of these results and also show that any symmetric convex body, which is sufficiently close to an unconditional body, satisfies the the reverse Blaschke-Santalo inequality. Archive classification: math.MG math.FA Mathematics Subject Classification: 52A20, 53A15, 52B10 Submitted from: jkim(a)math.kent.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1302.5719 or http://arXiv.org/abs/1302.5719

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Abstract of a paper by Claudia Correa and Daniel V. Tausk
by alspach＠math.okstate.edu 27 Feb '13

27 Feb '13

This is an announcement for the paper "Extension property and complementation of isometric copies of continuous functions spaces" by Claudia Correa and Daniel V. Tausk. Abstract: In this article we prove that every isometric copy of C(L) in C(K) is complemented if L is compact Hausdorff of finite height and K is a compact Hausdorff space satisfying the extension property, i.e., every closed subset of K admits an extension operator. The space C(L) can be replaced by its subspace C(L|F) consisting of functions that vanish on a closed subset F of L. In particular, we obtain that every isometric copy of c_0(I) in C(K) is complemented, if K has the extension property. Finally, we study the class of spaces having the extension property, establishing some closure results for this class and relating it to other classes of compact spaces. Archive classification: math.FA Mathematics Subject Classification: 46B20, 46E15, 54G12 Remarks: 9 pages Submitted from: tausk(a)ime.usp.br The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1302.4661 or http://arXiv.org/abs/1302.4661

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Abstract of a paper by Grzegorz Plebanek
by alspach＠math.okstate.edu 27 Feb '13

27 Feb '13

This is an announcement for the paper "On positive embeddings of C(K) spaces" by Grzegorz Plebanek. Abstract: We investigate isomorphic embeddings $T: C(K)\to C(L)$ between Banach spaces of continuous functions. We show that if such an embedding $T$ is a positive operator then $K$ is an image of $L$ under a upper semicontinuous set-function having finite values. Moreover we show that $K$ has a $\pi$-base of sets which closures a continuous images of compact subspaces of $L$. Our results imply in particular that if $C(K)$ can be positively embedded into $C(L)$ then some topological properties of $L$, such as countable tightness of Frechetness, pass to the space $K$. We show that some arbitrary isomorphic embeddings $C(K)\to C(L)$ can be, in a sense, reduced to positive embeddings. Archive classification: math.FA Submitted from: grzes(a)math.uni.wroc.pl The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1302.4360 or http://arXiv.org/abs/1302.4360

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Abstract of a paper by Grzegorz Plebanek
by alspach＠math.okstate.edu 27 Feb '13

27 Feb '13

This is an announcement for the paper "On isomorphisms of Banach spaces of continuous functions" by Grzegorz Plebanek. Abstract: We prove that if $K$ and $L$ are compact spaces and $C(K)$ and $C(L)$ are isomorphic as Banach spaces then $K$ has a $\pi$-base consisting of open sets $U$ such that $\overline{U}$ is a continuous image of some compact subspace of $L$. This gives some information on isomorphic classes of the spaces of the form $C([0,1]^\kappa)$ and $C(K)$ where $K$ is Corson compact. Archive classification: math.FA Mathematics Subject Classification: Primary 46B26, 46B03, 46E15 Remarks: 15 pages Submitted from: grzes(a)math.uni.wroc.pl The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1302.3211 or http://arXiv.org/abs/1302.3211

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Abstract of a paper by Miek Messerschmidt and Marcel de Jeu
by alspach＠math.okstate.edu 27 Feb '13

27 Feb '13

This is an announcement for the paper "Right inverses of surjections from cones onto Banach spaces" by Miek Messerschmidt and Marcel de Jeu. Abstract: Abstract. We show that a continuous additive positively homogeneous map from a closed not necessarily proper cone in a Banach space onto a Banach space is an open map precisely when it is surjective. This generalization of the usual Open Mapping Theorem for Banach spaces is then combined with Michael's Selection Theorem to yield the existence of a continuous bounded positively homogeneous right inverse of such a surjective map; an improved version of the usual Open Mapping Theorem is then a special case. As another consequence, a stronger version of the analogue of And\^o's Theorem for an ordered Banach space is obtained for a Banach space that is, more generally than in And\^o's Theorem, a sum of possibly uncountably many closed not necessarily proper cones. Applications are given for a (pre)-ordered Banach space and for various spaces of continuous functions taking values in such a Banach space or, more generally, taking values in an arbitrary Banach space that is a finite sum of closed not necessarily proper cones. Archive classification: math.FA Mathematics Subject Classification: Primary 47A05, Secondary 46A30, 46B20, 46B40 Submitted from: mmesserschmidt(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1302.2822 or http://arXiv.org/abs/1302.2822

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Abstract of a paper by Hu Bingyang, Le Hai Khoi, and Kehe Zhu
by alspach＠math.okstate.edu 27 Feb '13

27 Feb '13

This is an announcement for the paper "Frames and operators in Schatten classes" by Hu Bingyang, Le Hai Khoi, and Kehe Zhu. Abstract: Let $T$ be a compact operator on a separable Hilbert space $H$. We show that, for $2\le p<\infty$, $T$ belongs to the Schatten class $S_p$ if and only if $\{\|Tf_n\|\}\in \ell^p$ for \emph{every} frame $\{f_n\}$ in $H$; and for $0<p\le2$, $T$ belongs to $S_p$ if and only if $\{\|Tf_n\|\}\in\ell^p$ for \emph{some} frame $\{f_n\}$ in $H$. Similar conditions are also obtained in terms of the sequence $\{\langle Tf_n,f_n\rangle\}$ and the double-indexed sequence $\{\langle Tf_n,f_m\rangle\}$. Archive classification: math.FA Mathematics Subject Classification: 47B10, 46A35, 46B15 Remarks: 27 pages Submitted from: kzhu(a)math.albany.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1302.2490 or http://arXiv.org/abs/1302.2490

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Abstract of a paper by Stanislaw J. Szarek
by alspach＠math.okstate.edu 27 Feb '13

27 Feb '13

This is an announcement for the paper "On measures of symmetry and floating bodies" by Stanislaw J. Szarek. Abstract: We consider the following measure of symmetry of a convex n-dimensional body K: $\rho(K)$ is the smallest constant for which there is a point x in K such that for partitions of K by an n-1-dimensional hyperplane passing through x the ratio of the volumes of the two parts is at most $\rho(K)$. It is well known that $\rho(K)=1$ iff K is symmetric. We establish a precise upper bound on $\rho(K)$; this recovers a 1960 result of Grunbaum. We also provide a characterization of equality cases (relevant to recent results of Nill and Paffenholz about toric varieties) and relate these questions to the concept of convex floating bodies. Archive classification: math.MG math.FA Mathematics Subject Classification: 52A20, 52A40, 46B20 Remarks: 5 pages; this is a slightly edited manuscript from early '00s The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1302.2076 or http://arXiv.org/abs/1302.2076

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Banach