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.

我们介绍了内部包含零的凸体的张量积和在欧几里得空间中0对称凸体的张量积的概念。我们证明了0对称凸体的张量积与有限维空间上的张量范数之间存在双射。这种双射保留了对偶性，内射性和射影性。我们获得了0对称凸体的Grothendieck定理的公式，并用它给出了希尔伯特张量积的几何表示形式（最大为\（K_G \）-常数）。我们看到，这些张量积保持了具有足够对称性的性质，并表现出与Löwner和John椭球的关系。