November 2015 - Banach - mathdept.okstate.edu

Abstract of a paper by Miek Messerschmidt and Marten Wortel
by alspach＠math.okstate.edu 12 Nov '15

12 Nov '15

This is an announcement for the paper "The intrinsic metric on the unit sphere of a normed space" by Miek Messerschmidt and Marten Wortel. Abstract: Let $S$ denote the unit sphere of a real normed space. We show that the intrinsic metric on $S$ is strongly equivalent to the induced metric on $S$. Specifically, for all $x,y\in S$, \[ \|x-y\|\leq d(x,y)\leq\sqrt{2}\pi\|x-y\|, \] where $d$ denotes the intrinsic metric on $S$. Archive classification: math.FA math.MG Mathematics Subject Classification: Primary:46B10. Secondary: 51F99, 46B07 Submitted from: mmesserschmidt(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1510.07442 or http://arXiv.org/abs/1510.07442

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Abstract of a paper by Grigoris Paouris and Petros Valettas
by alspach＠math.okstate.edu 12 Nov '15

12 Nov '15

This is an announcement for the paper "On Dvoretzky's theorem for subspaces of $L_p$" by Grigoris Paouris and Petros Valettas. Abstract: We prove that for any $p > 2$ and every $n$-dimensional subspace $X$ of $L_p$, the Euclidean space $\ell_2^k$ can be $(1 + \varepsilon)$-embedded into $X$ with $k \geq c_p \min\{\varepsilon^2 n, (\varepsilon n)^{2/p} \}$, where $c_p > 0$ is a constant depending only on $p$. Archive classification: math.FA math.MG Mathematics Subject Classification: 46B07, 46B09 Remarks: 20 pages Submitted from: valettasp(a)missouri.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1510.07289 or http://arXiv.org/abs/1510.07289

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Abstract of a paper by Grigoris Paouris, Petros Valettas and Joel Zinn
by alspach＠math.okstate.edu 12 Nov '15

12 Nov '15

This is an announcement for the paper "Random version of Dvoretzky's theorem in $\ell_p^n$" by Grigoris Paouris, Petros Valettas and Joel Zinn. Abstract: We study the dependence on $\varepsilon$ in the critical dimension $k(n, p, \varepsilon)$ that one can find random sections of the $\ell_p^n$-ball which are $(1+\varepsilon)$-spherical. For any fixed $n$ we give lower estimates for $k(n, p, \varepsilon)$ for all eligible values $p$ and $\varepsilon$, which agree with the sharp estimates for the extreme values $p = 1$ and $p = \infty$. In order to do so, we provide bounds for the gaussian concentration of the $\ell_p$-norm. Archive classification: math.FA Mathematics Subject Classification: 46B06, 46B07, 46B09 Remarks: 45 pages Submitted from: valettasp(a)missouri.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1510.07284 or http://arXiv.org/abs/1510.07284

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