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

Abstract

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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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 24, 1373–1398 (2020). https://doi.org/10.1007/s11117-020-00736-y

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Keywords

  • 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