  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.
