Advertisement

The Tensorization Trick in Convex Geometry

  • Alexander BarvinokEmail author
Chapter
Part of the Bolyai Society Mathematical Studies book series (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.

Keywords

Tensor Convex body Approximation 

1991 Mathematics Subject Classification

15A69 52A20 52A45 52C17 14P05 

Notes

Acknowledgements

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

References

  1. 1.
    N. Alon, Problems and results in extremal combinatorics. I, EuroComb’01 (Barcelona). Discret. Math. 273(1–3), 31–53 (2003)CrossRefGoogle Scholar
  2. 2.
    N. Alon, Perturbed identity matrices have high rank: proof and applications. Comb. Probab. Comput. 18(1–2), 3–15 (2009)MathSciNetCrossRefGoogle Scholar
  3. 3.
    K. Ball, An elementary introduction to modern convex geometry, Flavors of Geometry, vol. 31, Mathematical Sciences Research Institute Publications (Cambridge University Press, Cambridge, 1997), pp. 1–58Google Scholar
  4. 4.
    I. Bárány, Z. Füredi, Approximation of the sphere by polytopes having few vertices. Proc. Am. Math. Soc. 102(3), 651–659 (1988)MathSciNetCrossRefGoogle Scholar
  5. 5.
    A. Barvinok, A Course in Convexity, vol. 54, Graduate Studies in Mathematics (American Mathematical Society, Providence, 2002)zbMATHGoogle Scholar
  6. 6.
    A. Barvinok, Estimating \(L^{\infty }\) norms by \(L^{2k}\) norms for functions on orbits. Found. Comput. Math. 2(4), 393–412 (2002)MathSciNetCrossRefGoogle Scholar
  7. 7.
    A. Barvinok, Approximating a norm by a polynomial, Geometric Aspects of Functional Analysis, vol. 1807, Lecture Notes in Mathematics (Springer, Berlin, 2003), pp. 20–26CrossRefGoogle Scholar
  8. 8.
    A. Barvinok, Thrifty approximations of convex bodies by polytopes. Int. Math. Res. Not. IMRN 2014(16), 4341–4356 (2014)MathSciNetCrossRefGoogle Scholar
  9. 9.
    A. Barvinok, G. Blekherman, Convex geometry of orbits, Combinatorial and Computational Geometry, vol. 52, Mathematical Sciences Research Institute Publications (Cambridge University Press, Cambridge, 2005), pp. 51–77Google Scholar
  10. 10.
    J. Batson, D.A. Spielman, N. Srivastava, Twice-Ramanujan sparsifiers. SIAM J. Comput. 41(6), 1704–1721 (2012)MathSciNetCrossRefGoogle Scholar
  11. 11.
    G. Blekherman, There are significantly more nonnegative polynomials than sums of squares. Isr. J. Math. 153, 355–380 (2006)MathSciNetCrossRefGoogle Scholar
  12. 12.
    K. Böröczky Jr., Approximation of general smooth convex bodies. Adv. Math. 153(2), 325–341 (2000)MathSciNetCrossRefGoogle Scholar
  13. 13.
    P. Borwein, T. Erdélyi, Polynomials and Polynomial Inequalities, vol. 161, Graduate Texts in Mathematics (Springer, New York, 1995)zbMATHGoogle Scholar
  14. 14.
    M. Braverman, K. Makarychev, Y. Makarychev, A. Naor, The Grothendieck constant is strictly smaller than Krivine’s bound. Forum of Mathematics Pi 1, e4–42 (2013)MathSciNetCrossRefGoogle Scholar
  15. 15.
    E.M. Bronshtein, Approximation of convex sets by polyhedra. Sovremennaya Matematika. Fundamental’nye Napravleniya 22, 5–37 (2007); translated in J. Math. Sci. (New York) 153(6), 727–762 (2008)Google Scholar
  16. 16.
    C. Carathéodory, Über den Variabilitatsbereich det Fourierschen Konstanten von Positiven harmonischen Furktionen. Rendiconti del Circolo Matematico di Palermo 32, 193–217 (1911)CrossRefGoogle Scholar
  17. 17.
    P. Frankl, R. Wilson, Intersection theorems with geometric consequences. Combinatorica 1, 357–368 (1981)MathSciNetCrossRefGoogle Scholar
  18. 18.
    D. Gale, Neighborly and cyclic polytopes, Proceedings of Symposia in Pure Mathematics, vol. VII (American Mathematical Society, Providence, 1963), pp. 225–232Google Scholar
  19. 19.
    E. Gluskin, A. Litvak, A remark on vertex index of the convex bodies, Geometric Aspects of Functional Analysis, vol. 2050, Lecture Notes in Mathematics (Springer, Heidelberg, 2012)CrossRefGoogle Scholar
  20. 20.
    A. Grothendieck, Résumé de la théorie métrique des produits tensoriels topologiques. Boletim da Sociedade Matemática São Paulo 8, 1–79 (1953)zbMATHGoogle Scholar
  21. 21.
    P.M. Gruber, Application of an idea of Voronoi to John type problems. Adv. Math. 218(2), 309–351 (2008)MathSciNetCrossRefGoogle Scholar
  22. 22.
    P.M. Gruber, Aspects of approximation of convex bodies, Handbook of Convex Geometry, vol. A, B (North-Holland, Amsterdam, 1993), pp. 319–345CrossRefGoogle Scholar
  23. 23.
    P.M. Gruber, Asymptotic estimates for best and stepwise approximation of convex bodies. I. Forum Math. 5(3), 281–297 (1993)MathSciNetzbMATHGoogle Scholar
  24. 24.
    L. Guth, N.H. Katz, On the Erdős distinct distances problem in the plane. Ann. Math. Second series 181(1), 155–190 (2015)MathSciNetCrossRefGoogle Scholar
  25. 25.
    F. John, Extremum problems with inequalities as subsidiary conditions, Studies and Essays, vol. 1948 (Interscience Publishers Inc., New York, 1948), pp. 187–204. Presented to R. Courant on his 60th Birthday, January 8Google Scholar
  26. 26.
    G.A. Kabatyanskii, V.I. Levenshtein, Bounds for packings on the sphere and in space (Russian). Problemy Peredachi Informacii 14(1), 3–25 (1978)MathSciNetzbMATHGoogle Scholar
  27. 27.
    J. Kahn, G. Kalai, A counterexample to Borsuk’s conjecture. Bull. Am. Math. Soc. New Series 29(1), 60–62 (1993)MathSciNetCrossRefGoogle Scholar
  28. 28.
    M. Kochol, Constructive approximation of a ball by polytopes. Mathematica Slovaca 44(1), 99–105 (1994)MathSciNetzbMATHGoogle Scholar
  29. 29.
    A.N. Kolmogorov, V.M. Tihomirov, \(\epsilon \)-entropy and \(\epsilon \)-capacity of sets in function spaces. Uspehi Matematicheskih Nauk 14(2(86)), 3–86 (1959). translated in Am. Math. Soc. Transl. 17(2), 277–364 (1961)Google Scholar
  30. 30.
    J.-L. Krivine, Constantes de Grothendieck et fonctions de type positif sur les sphères. Adv. Math. 31(1), 16–30 (1979)MathSciNetCrossRefGoogle Scholar
  31. 31.
    A. Litvak, M. Rudelson, N. Tomczak-Jaegermann, On approximation by projections of polytopes with few facets. Isr. J. Math. 203(1), 141–160 (2014)MathSciNetCrossRefGoogle Scholar
  32. 32.
    J. Matoušek, Thirty-Three Miniatures. Mathematical and Algorithmic Applications of Linear Algebra, vol. 53, Student Mathematical Library (American Mathematical Society, Providence, 2010)zbMATHGoogle Scholar
  33. 33.
    P. Parrilo, A. Jadbabaie, Approximation of the joint spectral radius using sum of squares. Linear Algebra Appl. 428(10), 2385–2402 (2008)MathSciNetCrossRefGoogle Scholar
  34. 34.
    G. Pisier, The Volume of Convex Bodies and Banach Space Geometry, vol. 94, Cambridge Tracts in Mathematics (Cambridge University Press, Cambridge, 1989)CrossRefGoogle Scholar
  35. 35.
    A.H. Stone, J.W. Tukey, Generalized “sandwich” theorems. Duke Math. J. 9, 356–359 (1942)MathSciNetCrossRefGoogle Scholar
  36. 36.
    T. Tao, A cheap version of the Kabatjanskii-Levenstein bound for almost orthogonal vectors (2013), blog post at https://terrytao.wordpress.com/2013/07/18/

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

Personalised recommendations