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New Trends in Intuitive Geometry pp 1–23Cite as

The Tensorization Trick in Convex Geometry

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Part of the book series: Bolyai Society Mathematical Studies ((BSMS,volume 27))

Abstract

The “tensorization trick” consists in proving some geometric result for a set of vectors \(\left\{ v_i\right\} \) in some vector space V and then applying the same result to the tensor powers \(\left\{ v_i^{\otimes k}\right\} \) in \(V^{\otimes k}\), which in turn produces a considerably stronger version of the original result for vectors \(\left\{ v_i\right\} \). 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 \({\mathbb R}^d\) and to constructing a counterexample to Borsuk’s conjecture.

This research was partially supported by NSF Grant DMS 1361541.

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Acknowledgements

I am grateful to Terence Tao for pointing to [1, 2, 36].

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Authors and Affiliations

  1. Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109-1043, USA

    Alexander Barvinok

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  1. Alexander Barvinok
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Correspondence to Alexander Barvinok .

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Editors and Affiliations

  1. MTA Alfréd Rényi Institute of Mathematics, Budapest, Hungary

    Gergely Ambrus

  2. MTA Alfréd Rényi Institute of Mathematics, Budapest, Hungary

    Imre Bárány

  3. MTA Alfréd Rényi Institute of Mathematics, Budapest, Hungary

    Károly J. Böröczky

  4. MTA Alfréd Rényi Institute of Mathematics, Budapest, Hungary

    Gábor Fejes Tóth

  5. MTA Alfréd Rényi Institute of Mathematics, Budapest, Hungary

    János Pach

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Barvinok, A. (2018). The Tensorization Trick in Convex Geometry. In: Ambrus, G., Bárány, I., Böröczky, K., Fejes Tóth, G., Pach, J. (eds) New Trends in Intuitive Geometry. Bolyai Society Mathematical Studies, vol 27. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-57413-3_1

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