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

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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 (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