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New Trends in Intuitive GeometryThe Tensorization Trick in Convex Geometry

New Trends in Intuitive Geometry: The Tensorization Trick in Convex Geometry [The “tensorization trick” consists in proving some geometric result for a set of vectors \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{ v_i\right\} $$\end{document} in some vector space V and then applying the same result to the tensor powers \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{ v_i^{\otimes k}\right\} $$\end{document} in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V^{\otimes k}$$\end{document}, which in turn produces a considerably stronger version of the original result for vectors \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{ v_i\right\} $$\end{document}. Our main examples concern packing vectors in the sphere, approximation of convex bodies by algebraic hypersurfaces and approximation of convex bodies by polytopes. We also discuss applications of a closely related polynomial method to constructing neighborly polytopes, bounding the Grothendieck constant, proving the polynomial ham sandwich theorem, bounding the number of equiangular lines in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb R}^d$$\end{document} and to constructing a counterexample to Borsuk’s conjecture.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

New Trends in Intuitive GeometryThe Tensorization Trick in Convex Geometry

Part of the Bolyai Society Mathematical Studies Book Series (volume 27)
Editors: Ambrus, Gergely; Bárány, Imre; Böröczky, Károly J.; Fejes Tóth, Gábor ; Pach, János
Barvinok, Alexander
New Trends in Intuitive Geometry — Nov 3, 2018
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22 pages

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Publisher
Springer Berlin Heidelberg
Copyright
© János Bolyai Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2018. Corrected Publication 2018
ISBN
978-3-662-57412-6
Pages
1 –23
DOI
10.1007/978-3-662-57413-3_1
Publisher site
See Chapter on Publisher Site

Abstract

[The “tensorization trick” consists in proving some geometric result for a set of vectors \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{ v_i\right\} $$\end{document} in some vector space V and then applying the same result to the tensor powers \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{ v_i^{\otimes k}\right\} $$\end{document} in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V^{\otimes k}$$\end{document}, which in turn produces a considerably stronger version of the original result for vectors \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{ v_i\right\} $$\end{document}. Our main examples concern packing vectors in the sphere, approximation of convex bodies by algebraic hypersurfaces and approximation of convex bodies by polytopes. We also discuss applications of a closely related polynomial method to constructing neighborly polytopes, bounding the Grothendieck constant, proving the polynomial ham sandwich theorem, bounding the number of equiangular lines in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb R}^d$$\end{document} and to constructing a counterexample to Borsuk’s conjecture.]

Published: Nov 3, 2018

Keywords: Tensor; Convex body; Approximation; 15A69; 52A20; 52A45; 52C17; 14P05

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