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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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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DOI: https://doi.org/10.1007/s11117-020-00736-y
Keywords
- Convex body
- Tensor product of convex sets
- Tensor product of banach spaces
- Hilbertian tensor norm
- Ideals of linear operators
- Grothendieck’s inequality