{ "id": "1906.11377", "version": "v1", "published": "2019-06-26T22:58:38.000Z", "updated": "2019-06-26T22:58:38.000Z", "title": "A General Theory of Tensor Products of Convex Sets in Euclidean Spaces", "authors": [ "Maite Fernández-Unzueta", "Luisa F. Higueras-Montaño" ], "comment": "21 pages", "categories": [ "math.FA" ], "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 L\\\"owner and the John ellipsoids.", "revisions": [ { "version": "v1", "updated": "2019-06-26T22:58:38.000Z" } ], "analyses": { "subjects": [ "46M05", "52A21", "47L20", "15A69" ], "keywords": [ "euclidean spaces", "symmetric convex bodies", "convex sets", "general theory", "bijection preserves duality" ], "note": { "typesetting": "TeX", "pages": 21, "language": "en", "license": "arXiv", "status": "editable" } } }