# A General Theory of Tensor Products of Convex Sets in Euclidean Spaces

## TL;DR

This paper develops a unified geometric framework for tensor products of convex sets in Euclidean spaces, establishing a correspondence with tensor norms and extending classical theorems like Grothendieck's.

## Contribution

It introduces tensor products of convex bodies, links them to tensor norms, and generalizes Grothendieck's theorem in a geometric setting.

## Key findings

- Bijective correspondence between tensor products of convex bodies and tensor norms.
- Formulation of Grothendieck's theorem for convex bodies.
- Geometric representation of Hilbertian tensor product.

## 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.