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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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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