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

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 s...

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Published inPositivity : an international journal devoted to the theory and applications of positivity in analysis Vol. 24; no. 5; pp. 1373 - 1398
Main Authors Fernández-Unzueta, Maite, Higueras-Montaño, Luisa F.
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 01.11.2020
Springer Nature B.V
Subjects
Calculus of Variations and Optimal Control; Optimization
Convexity
Econometrics
Ellipsoids
Euclidean geometry
Euclidean space
Fourier Analysis
Mathematical analysis
Mathematics
Mathematics and Statistics
Norms
Operator Theory
Potential Theory
Tensors
Tensor product of convex sets
52A21
47L20
Tensor product of banach spaces
Hilbertian tensor norm
Ideals of linear operators
46M05
Convex body
Grothendieck’s inequality
15A69
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Summary: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.
ISSN:1385-1292
1572-9281
DOI:10.1007/s11117-020-00736-y