The Tensorization Trick in Convex Geometry

  • Alexander BarvinokEmail author
Part of the Bolyai Society Mathematical Studies book series (BSMS, volume 27)


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.


Tensor Convex body Approximation 

1991 Mathematics Subject Classification

15A69 52A20 52A45 52C17 14P05 



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


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Copyright information

© János Bolyai Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MichiganAnn ArborUSA

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