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Abstract of a paper by Ivan Feshchenko
by alspach＠math.okstate.edu 04 Sep '12

04 Sep '12

This is an announcement for the paper "On the divergence of series of the form sum_{k=1}^infty||A_k x||^p" by Ivan Feshchenko. Abstract: Let {A} be a system of operators. With any element x we associate the set of elements {Ax}. We study conditions under which there exists an element x such that the sum of p-th powers of norms of the elements {Ax} is equal to infinity. Archive classification: math.FA Mathematics Subject Classification: 40H05, 46B20, 47A05 Remarks: 9 pages, submitted to Studia Mathematica Submitted from: ivanmath007(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1208.1863 or http://arXiv.org/abs/1208.1863

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Abstract of a paper by Alina Stancu
by alspach＠math.okstate.edu 25 Aug '12

25 Aug '12

This is an announcement for the paper "Some affine invariants revisited" by Alina Stancu. Abstract: We present several sharp inequalities for the $SL(n)$ invariant $\Omega_{2,n}(K)$ introduced in our earlier work on centro-affine invariants for smooth convex bodies containing the origin. A connection arose with the Paouris-Werner invariant $\Omega_K$ defined for convex bodies $K$ whose centroid is at the origin. We offer two alternative definitions for $\Omega_K$ when $K \in C^2_+$. The technique employed prompts us to conjecture that any $SL(n)$ invariant of convex bodies with continuous and positive centro-affine curvature function can be obtained as a limit of normalized $p$-affine surface areas of the convex body. Archive classification: math.FA Mathematics Subject Classification: 52A40, 52A38 Remarks: 15 pages Submitted from: stancu(a)mathstat.concordia.ca The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1208.0783 or http://arXiv.org/abs/1208.0783

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Abstract of a paper by Jop Briet, Assaf Naor, and Oded Regev
by alspach＠math.okstate.edu 25 Aug '12

25 Aug '12

This is an announcement for the paper "Locally decodable codes and the failure of cotype for projective tensor products" by Jop Briet, Assaf Naor, and Oded Regev. Abstract: It is shown that for every $p\in (1,\infty)$ there exists a Banach space $X$ of finite cotype such that the projective tensor product $\ell_p\tp X$ fails to have finite cotype. More generally, if $p_1,p_2,p_3\in (1,\infty)$ satisfy $\frac{1}{p_1}+\frac{1}{p_2}+\frac{1}{p_3}\le 1$ then $\ell_{p_1}\tp\ell_{p_2}\tp\ell_{p_3}$ does not have finite cotype. This is a proved via a connection to the theory of locally decodable codes. Archive classification: math.FA cs.CC Submitted from: odedr(a)cs.tau.ac.il The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1208.0539 or http://arXiv.org/abs/1208.0539

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Abstract of a paper by Rolf Schneider and Franz E. Schuster
by alspach＠math.okstate.edu 25 Aug '12

25 Aug '12

This is an announcement for the paper "Rotation equivariant Minkowski valuations" by Rolf Schneider and Franz E. Schuster. Abstract: The projection body operator \Pi, which associates with every convex body in Euclidean space Rn its projection body, is a continuous valuation, it is invariant under translations and equivariant under rotations. It is also well known that \Pi\ maps the set of polytopes in Rn into itself. We show that \Pi\ is the only non-trivial operator with these properties. Archive classification: math.MG math.DG math.FA Mathematics Subject Classification: 52A20, 52B11, 52B45 Citation: Int. Math. Res. Not. 2006, Art. ID 72894, 20 pp Submitted from: franz.schuster(a)tuwien.ac.at The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1207.7279 or http://arXiv.org/abs/1207.7279

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Abstract of a paper by Lukas Parapatits and Franz E. Schuster
by alspach＠math.okstate.edu 25 Aug '12

25 Aug '12

This is an announcement for the paper "The Steiner formula for Minkowski faluations" by Lukas Parapatits and Franz E. Schuster. Abstract: A Steiner type formula for continuous translation invariant Minkowski valuations is established. In combination with a recent result on the symmetry of rigid motion invariant homogeneous bivaluations, this new Steiner type formula is used to obtain a family of Brunn-Minkowski type inequalities for rigid motion intertwining Minkowski valuations. Archive classification: math.MG math.DG math.FA Mathematics Subject Classification: 52B45, 52A40 Citation: Adv. in Math. 230 (2012), 978-994 Submitted from: franz.schuster(a)tuwien.ac.at The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1207.7276 or http://arXiv.org/abs/1207.7276

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Abstract of a paper by Gabriel Maresch and Franz E. Schuster
by alspach＠math.okstate.edu 25 Aug '12

25 Aug '12

This is an announcement for the paper "The Sine transform of isotropic measures" by Gabriel Maresch and Franz E. Schuster. Abstract: Sharp isoperimetric inequalities for the sine transform of even isotropic measures are established. The corresponding reverse inequalities are obtained in an asymptotically optimal form. These new inequalities have direct applications to strong volume estimates for convex bodies from data about their sections or projections. Archive classification: math.MG math.DG math.FA Mathematics Subject Classification: 52A20, 52A41, 53Cxx Citation: Int. Math. Res. Not. IMRN 2012, 717–739 Submitted from: franz.schuster(a)tuwien.ac.at The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1207.7266 or http://arXiv.org/abs/1207.7266

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Abstract of a paper by Franz E. Schuster and Thomas Wannerer
by alspach＠math.okstate.edu 25 Aug '12

25 Aug '12

This is an announcement for the paper "GL(n) contravariant Minkowski valuations" by Franz E. Schuster and Thomas Wannerer. Abstract: A complete classification of all continuous GL(n) contravariant Minkowski valuations is established. As an application we present a family of sharp isoperimetric inequalities for such valuations which generalize the classical Petty projection inequality. Archive classification: math.MG math.DG math.FA Mathematics Subject Classification: 52B45, 52A20, 52A40 Citation: Trans. Amer. Math. Soc. 364 (2012), 815–826 Submitted from: franz.schuster(a)tuwien.ac.at The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1207.7256 or http://arXiv.org/abs/1207.7256

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Abstract of a paper by Franz E. Schuster
by alspach＠math.okstate.edu 25 Aug '12

25 Aug '12

This is an announcement for the paper "Crofton measures and Minkowski valuations" by Franz E. Schuster. Abstract: A description of continuous rigid motion compatible Minkowski valuations is established. As an application, we present a Brunn-Minkowski type inequality for intrinsic volumes of these valuations. Archive classification: math.MG math.DG math.FA Mathematics Subject Classification: 52B45, 43A90, 52A40 Citation: Duke Math. J. 154 (2010), 1–30 Submitted from: franz.schuster(a)tuwien.ac.at The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1207.7254 or http://arXiv.org/abs/1207.7254

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Abstract of a paper by Franz E. Schuster
by alspach＠math.okstate.edu 25 Aug '12

25 Aug '12

This is an announcement for the paper "Convolutions and multiplier transformations of convex bodies" by Franz E. Schuster. Abstract: Rotation intertwining maps from the set of convex bodies in Rn into itself that are continuous linear operators with respect to Minkowski and Blaschke addition are investigated. The main focus is on Blaschke-Minkowski homomorphisms. We show that such maps are represented by a spherical convolution operator. An application of this representation is a complete classification of all even Blaschke-Minkowski homomorphisms which shows that these maps behave in many respects similar to the well known projection body operator. Among further applications is the following result: If an even Blaschke-Minkowski homomorphism maps a convex body to a polytope, then it is a constant multiple of the projection body operator. Archive classification: math.MG math.DG math.FA Mathematics Subject Classification: 52A20, 43A90, 52A40 Citation: Trans. Amer. Math. Soc. 359 (2007), 5567–5591 Submitted from: franz.schuster(a)tuwien.ac.at The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1207.7252 or http://arXiv.org/abs/1207.7252

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Abstract of a paper by Peter M. Gruber and Franz E. Schuster
by alspach＠math.okstate.edu 25 Aug '12

25 Aug '12

This is an announcement for the paper "An arithmetic proof of John’s ellipsoid theorem" by Peter M. Gruber and Franz E. Schuster. Abstract: Using an idea of Voronoi in the geometric theory of positive definite quadratic forms, we give a transparent proof of John’s characterization of the unique ellipsoid of maximum volume contained in a convex body. The same idea applies to the ‘hard part’ of a generalization of John’s theorem and shows the difficulties of the corresponding ‘easy part’. Archive classification: math.MG math.DG math.FA Mathematics Subject Classification: 52A21, 46B07, 52A27 Citation: Arch. Math. (Basel) 85 (2005), 82–88 Submitted from: franz.schuster(a)tuwien.ac.at The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1207.7246 or http://arXiv.org/abs/1207.7246

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