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A general theory of tensor products of convex sets in Euclidean spaces

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We introduce both the notions of tensor product of convex bodies that contain zero in the interior, and of tensor product of 0-symmetric convex bodies in Euclidean spaces. We prove that there is a bijection between tensor products of 0-symmetric convex bodies and tensor norms on finite dimensional spaces. This bijection preserves duality, injectivity and projectivity. We obtain a formulation of Grothendieck‘s Theorem for 0-symmetric convex bodies and use it to give a geometric representation (up to the \(K_G\)-constant) of the Hilbertian tensor product. We see that the property of having enough symmetries is preserved by these tensor products, and exhibit relations with the Löwner and the John ellipsoids.

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  1. 1.

    Artstein-Avidan, S., Giannopoulos, A., Milman, V.D.: Asymptotic Geometric Analysis. Part I. American Mathematical Society, Providence, RI (2015)

  2. 2.

    Aubrun, G., Szarek, S.: Tensor products of convex sets and the volume of separable states on n qudits. Phys. Rev. A 73(2), 022109 (2006)

  3. 3.

    Aubrun, G., Szarek, S.: Alice and Bob Meet Banach: The Interface of Asymptotic Geometric Analysis and Quantum Information Theory. American Mathematical Soc. (2017)

  4. 4.

    Behrends, E., Wittstock, G.: Tensorprodukte und simplexe. Invent. Math. 11, 188–198 (1970)

  5. 5.

    Davies, E., Vincent-Smith, G.: Tensor products, infinite products, and projective limits of Choquet simplexes. Math. Scand. 22(1), 145–164 (1969)

  6. 6.

    Defant, A., Floret, K.: Tensor Norms and Operator Ideals. North Holland Mathematics Studies, Amsterdam (1992)

  7. 7.

    Diestel, J., Fourie, J., Swart, J.: The Metric Theory of Tensor Products. American Mathematical Society, Providence, RI (2008). Grothendieck’s résumé revisited

  8. 8.

    Diestel, J., Jarchow, H., Tonge, A.: Absolutely Summing Operators. Cambridge University Press, Cambridge (1995)

  9. 9.

    Efremenko, K.: 3-query locally decodable codes of subexponential length. In: STOC’09-Proceedings of the 2009 ACM International Symposium on Theory of Computing, pp. 39–44 (2009)

  10. 10.

    Fernández-Unzueta, M., Higueras-Montaño, L.: Convex bodies associated to tensor norms. J. Convex Anal. 26, 4 (2019)

  11. 11.

    Grothendieck, A.: Résumé de la théorie métrique des produits tensoriels topologiques. Bol. Soc. Mat. São Paulo 8, 1–79 (1953)

  12. 12.

    Hulanicki, A., Phelps, R.: Some applications of tensor products of partially-ordered linear spaces. J. Funct. Anal. 2, 177–201 (1968)

  13. 13.

    John, F.: Extremum problems with inequalities as subsidiary conditions. Studies and Essays Presented to R. Courant on his 60th Birthday, pp. 187–204. Interscience Publishers Inc, New York (1948)

  14. 14.

    Johnson, W., Lindenstrauss, J.: Basic concepts in the geometry of banach spaces. In: Johnson, W., Lindenstrauss, J. (eds.) Handbook of the Geometry of Banach Spaces, vol. 1, pp. 1–84. Elsevier, Amsterdam (2001)

  15. 15.

    Kadison, R., Ringrose, J.: Fundamentals of the Theory of Operator Algebras. Academic Press, Edinburgh (1983)

  16. 16.

    Kakutani, S.: Some characterizations of Euclidean space. Jap. J. Math. 16, 93–97 (1939)

  17. 17.

    Khot, S., Naor, A.: Grothendieck-type inequalities in combinatorial optimization. Commun. Pure Appl. Math. 65(7), 992–1035 (2012)

  18. 18.

    Lazar, A.: Affine products of simplexes. Math. Scand. 22, 165–175 (1968)

  19. 19.

    Minkowski, H.: Geometrie der Zahlen 1910. Teubner, Leipzig (1927)

  20. 20.

    Namioka, I., Phelps, R.: Tensor products of compact convex sets. Pac. J. Math. 31(2), 469–480 (1969)

  21. 21.

    Pietsch, A.: Operator Ideals. North-Holland Publishing Company, Amsterdam (1980)

  22. 22.

    Pisier, G.: Factorization of linear operators and Geometry of Banach spaces. In: Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI (1986)

  23. 23.

    Pisier, G.: Grothendieck’s theorem, past and present. Bull. Am. Math. Soc. 49(2), 237–323 (2012)

  24. 24.

    Ryan, R.: Introduction to Tensor Products of Banach Spaces. Springer, Berlin (2002)

  25. 25.

    Schneider, R.: Convex bodies: the Brunn-Minkowski theory. Cambridge University Press, Cambridge (1993)

  26. 26.

    Semadeni, Z.: Categorical methods in convexity. In: Proceedings of Colloquium on Convexity (Copenhagen, 1965), pp. 281–307. Kobenhavns Univ. Mat. Inst., Copenhagen (1967)

  27. 27.

    Tomczak-Jaegermann, N.: Banach-Mazur Distances and Finite-Dimensional Operator Ideals. Longman Sc & Tech, Harlow (1989)

  28. 28.

    Velasco, M.: Linearization functors on real convex sets. SIAM J. Optim. 25(1), 1–27 (2015)

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Correspondence to Luisa F. Higueras-Montaño.

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The first author was partially supported by Consejo Nacional de Ciencia y Tecnología (CONACyT), grant number 284110. The second named author was supported by CONACyT scholarship for Ph.D. studies.

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Fernández-Unzueta, M., Higueras-Montaño, L.F. A general theory of tensor products of convex sets in Euclidean spaces. Positivity (2020).

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  • Convex body
  • Tensor product of convex sets
  • Tensor product of banach spaces
  • Hilbertian tensor norm
  • Ideals of linear operators
  • Grothendieck’s inequality

Mathematics Subject Classification

  • 46M05
  • 52A21
  • 47L20
  • 15A69