Banach

banach@mathdept.okstate.edu

2549 discussions

Abstract of a paper by Dario Cordero-Erausquin and Boaz Klartag
by alspach＠math.okstate.edu 29 Sep '11

29 Sep '11

This is an announcement for the paper "Interpolations, convexity and geometric inequalities" by Dario Cordero-Erausquin and Boaz Klartag. Abstract: We survey some interplays between spectral estimates of H\"ormander-type, degenerate Monge-Amp\`ere equations and geometric inequalities related to log-concavity such as Brunn-Minkowski, Santal\'o or Busemann inequalities. Archive classification: math.FA math.CV Submitted from: cordero(a)math.jussieu.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1109.3652 or http://arXiv.org/abs/1109.3652

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Abstract of a paper by Gilles Pisier
by alspach＠math.okstate.edu 29 Sep '11

29 Sep '11

This is an announcement for the paper "Real interpolation between row and column spaces" by Gilles Pisier. Abstract: We give an equivalent expression for the $K$-functional associated to the pair of operator spaces $(R,C)$ formed by the rows and columns respectively. This yields a description of the real interpolation spaces for the pair $(M_n(R), M_n(C))$ (uniformly over $n$). More generally, the same result is valid when $M_n$ (or $B(\ell_2)$) is replaced by any semi-finite von~Neumann algebra. We prove a version of the non-commutative Khintchine inequalities (originally due to Lust--Piquard) that is valid for the Lorentz spaces $L_{p,q}(\tau)$ associated to a non-commutative measure $\tau$, simultaneously for the whole range $1\le p,q< \infty$, regardless whether $p<2 $ or $p>2$. Actually, the main novelty is the case $p=2,q\not=2$. We also prove a certain simultaneous decomposition property for the operator norm and the Hilbert-Schmidt one. Archive classification: math.OA Mathematics Subject Classification: 47B10 Submitted from: pisier(a)math.jussieu.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1109.1860 or http://arXiv.org/abs/1109.1860

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Abstract of a paper by Alistair Bird, Graham Jameson and Niels Jakob Laustsen
by alspach＠math.okstate.edu 29 Sep '11

29 Sep '11

This is an announcement for the paper "The Giesy--James theorem for general index $p$, with an application to operator ideals on the $p$th James space" by Alistair Bird, Graham Jameson and Niels Jakob Laustsen. Abstract: A theorem of Giesy and James states that $c_0$ is finitely representable in James' quasi-reflexive Banach space $J_2$. We extend this theorem to the $p$th quasi-reflexive James space $J_p$ for each $p \in (1,\infty)$. As an application, we obtain a new closed ideal of operators on $J_p$, namely the closure of the set of operators that factor through the complemented subspace $(\ell_\infty^1 \oplus \ell_\infty^2 \oplus \cdots \oplus \ell_\infty^n \oplus \cdots)_{\ell_p}$ of $J_p$. Archive classification: math.FA math.OA Mathematics Subject Classification: 46B45, 47L20 (Primary) 46B07, 46H10, 47L10 (Secondary) Remarks: 16 pages Submitted from: alistairbird(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1109.1776 or http://arXiv.org/abs/1109.1776

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Abstract of a paper by Gilles Pisier
by alspach＠math.okstate.edu 29 Sep '11

29 Sep '11

This is an announcement for the paper "Real interpolation and transposition of certain function spaces" by Gilles Pisier. Abstract: Our starting point is a lemma due to Varopoulos. We give a different proof of a generalized form this lemma, that yields an equivalent description of the $K$-functional for the interpolation couple $(X_0,X_1)$ where $X_0=L_{p_0,\infty}(\mu_1; L_q(\mu_2))$ and $X_1=L_{p_1,\infty}(\mu_2; L_q(\mu_1))$ where $0<q<p_0,p_1\le \infty$ and $(\Omega_1,\mu_1), (\Omega_2,\mu_2)$ are arbitrary measure spaces. When $q=1$, this implies that the space $(X_0,X_1)_{\theta,\infty}$ ($0<\theta<1$) can be identified with a certain space of operators. We also give an extension of the Varopoulos Lemma to pairs (or finite families) of conditional expectations that seems of independent interest. The present paper is motivated by non-commutative applications that we choose to publish separately. Archive classification: math.FA Submitted from: pisier(a)math.jussieu.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1109.1006 or http://arXiv.org/abs/1109.1006

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Abstract of a paper by Anna Kaminska, Alexey I. Popov, Eugeniu Spinu, Adi Tcaciuc, and Vladimir G. Troitsky
by alspach＠math.okstate.edu 09 Sep '11

09 Sep '11

This is an announcement for the paper "Norm closed operator ideals in Lorentz sequence spaces" by Anna Kaminska, Alexey I. Popov, Eugeniu Spinu, Adi Tcaciuc, and Vladimir G. Troitsky. Abstract: In this paper, we study the structure of closed algebraic ideals in the algebra of operators acting on a Lorentz sequence space. Archive classification: math.FA Mathematics Subject Classification: Primary: 47L20. Secondary: 47B10, 47B37 Remarks: 25 pages Submitted from: troitsky(a)ualberta.ca The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1108.6026 or http://arXiv.org/abs/1108.6026

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Abstract of a paper by Almut Burchard
by alspach＠math.okstate.edu 09 Sep '11

09 Sep '11

This is an announcement for the paper "Rate of convergence of random polarizations" by Almut Burchard. Abstract: After n random polarizations of Borel set on a sphere, its expected symmetric difference from a polar cap is bounded by C/n, where the constant depends on the dimension [arXiv:1104.4103]. We show here that this power law is best possible, and that the constant grows at least linearly with the dimension. Archive classification: math.PR math.FA Remarks: 5 pages 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/1108.5500 or http://arXiv.org/abs/1108.5500

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Abstract of a paper by Horst Martini, Konrad J. Swanepoel, and P. Oloff de Wet
by alspach＠math.okstate.edu 09 Sep '11

09 Sep '11

This is an announcement for the paper "Absorbing angles, Steiner minimal trees, and antipodality" by Horst Martini, Konrad J. Swanepoel, and P. Oloff de Wet. Abstract: We give a new proof that a star $\{op_i:i=1,\dots,k\}$ in a normed plane is a Steiner minimal tree of its vertices $\{o,p_1,\dots,p_k\}$ if and only if all angles formed by the edges at $o$ are absorbing [Swanepoel, Networks \textbf{36} (2000), 104--113]. The proof is more conceptual and simpler than the original one. We also find a new sufficient condition for higher-dimensional normed spaces to share this characterization. In particular, a star $\{op_i: i=1,\dots,k\}$ in any CL-space is a Steiner minimal tree of its vertices $\{o,p_1,\dots,p_k\}$ if and only if all angles are absorbing, which in turn holds if and only if all distances between the normalizations $\frac{1}{\|p_i\|}p_i$ equal $2$. CL-spaces include the mixed $\ell_1$ and $\ell_\infty$ sum of finitely many copies of $R^1$. Archive classification: math.MG math.FA Mathematics Subject Classification: 46B20 (Primary). 05C05, 49Q10, 52A21 (Secondary) Citation: Journal of Optimization Theory and Applications, 143 (2009), The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1108.5046 or http://arXiv.org/abs/1108.5046

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Abstract of a paper by Boaz Klartag and Emanuel Milman
by alspach＠math.okstate.edu 09 Sep '11

09 Sep '11

This is an announcement for the paper "Inner regularization of log-concave measures and small-ball estimates" by Boaz Klartag and Emanuel Milman. Authors: Bo'az Klartag and Emanuel Milman Abstract: In the study of concentration properties of isotropic log-concave measures, it is often useful to first ensure that the measure has super-Gaussian marginals. To this end, a standard preprocessing step is to convolve with a Gaussian measure, but this has the disadvantage of destroying small-ball information. We propose an alternative preprocessing step for making the measure seem super-Gaussian, at least up to reasonably high moments, which does not suffer from this caveat: namely, convolving the measure with a random orthogonal image of itself. As an application of this ``inner-thickening", we recover Paouris' small-ball estimates. Archive classification: math.FA Remarks: 12 pages Submitted from: emanuel.milman(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1108.4856 or http://arXiv.org/abs/1108.4856

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Abstract of a paper by Emanuel Milman
by alspach＠math.okstate.edu 09 Sep '11

09 Sep '11

This is an announcement for the paper "Sharp isoperimetric inequalities and model spaces for curvature-dimension-diameter condition" by Emanuel Milman. Abstract: We obtain new sharp isoperimetric inequalities on a Riemannian manifold equipped with a probability measure, whose generalized Ricci curvature is bounded from below (possibly negatively), and generalized dimension and diameter of the convex support are bounded from above (possibly infinitely). Our inequalities are \emph{sharp} for sets of any given measure and with respect to all parameters (curvature, dimension and diameter). Moreover, for each choice of parameters, we identify the \emph{model spaces} which are extremal for the isoperimetric problem. In particular, we recover the Gromov--L\'evy and Bakry--Ledoux isoperimetric inequalities, which state that whenever the curvature is strictly \emph{positively} bounded from below, these model spaces are the $n$-sphere and Gauss space, corresponding to generalized dimension being $n$ and $\infty$, respectively. In all other cases, which seem new even for the classical Riemannian-volume measure, it turns out that there is no \emph{single} model space to compare to, and that a simultaneous comparison to a natural \emph{one parameter family} of model spaces is required, nevertheless yielding a sharp result. Archive classification: math.DG math.FA math.MG Mathematics Subject Classification: 32F32, 53C21, 53C20 Remarks: 36 pages Submitted from: emanuel.milman(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/mod/304481 or http://arXiv.org/abs/mod/304481

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Abstract of a paper by Veronica Dimant, Daniel Galicer and Ricardo Garcia
by alspach＠math.okstate.edu 09 Sep '11

09 Sep '11

This is an announcement for the paper "Geometry of integral polynomials, $M$-ideals and unique norm preserving extensions" by Veronica Dimant, Daniel Galicer and Ricardo Garcia. Abstract: We use the Aron-Berner extension to prove that the set of extreme points of the unit ball of the space of integral polynomials over a real Banach space $X$ is $\{\pm \phi^k: \phi \in X^*, \| \phi\|=1\}$. With this description we show that, for real Banach spaces $X$ and $Y$, if $X$ is a non trivial $M$-ideal in $Y$, then $\widehat\bigotimes^{k,s}_{\varepsilon_{k,s}} X$ (the $k$-th symmetric tensor product of $X$ endowed with the injective symmetric tensor norm) is \emph{never} an $M$-ideal in $\widehat\bigotimes^{k,s}_{\varepsilon_{k,s}} Y$. This result marks up a difference with the behavior of non-symmetric tensors since, when $X$ is an $M$-ideal in $Y$, it is known that $\widehat\bigotimes^k_{\varepsilon_k} X$ (the $k$-th tensor product of $X$ endowed with the injective tensor norm) is an $M$-ideal in $\widehat\bigotimes^k_{\varepsilon_k} Y$. Nevertheless, if $X$ is Asplund, we prove that every integral $k$-homogeneous polynomial in $X$ has a unique extension to $Y$ that preserves the integral norm. We explicitly describe this extension. We also give necessary and sufficient conditions (related with the continuity of the Aron-Berner extension morphism) for a fixed $k$-homogeneous polynomial $P$ belonging to a maximal polynomial ideal $\Q(^kX)$ to have a unique norm preserving extension to $\Q(^kX^{**})$. To this end, we study the relationship between the bidual of the symmetric tensor product of a Banach space and the symmetric tensor product of its bidual and show (in the presence of the BAP) that both spaces have `the same local structure'. Other applications to the metric and isomorphic theory of symmetric tensor products and polynomial ideals are also given. Archive classification: math.FA Mathematics Subject Classification: 46G25, 46M05, 46B28 Remarks: 25 pages Submitted from: dgalicer(a)dm.uba.ar The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1108.3975 or http://arXiv.org/abs/1108.3975

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