A geometric approach to tensor products of convex sets via tensor norms

The talk, given by Luisa Fernanda Higueras at a conference honoring Vilker Bach’s 60th birthday, presents a geometric approach to defining tensor products of convex sets, inspired by the theory of tensor norms on Banach spaces. The speaker begins by reviewing basic concepts from tensor norms and convex geometry, including reasonable cross norms, projective and injective tensor norms, and the correspondence between origin-symmetric convex bodies and norms via the Minkowski functional. She then introduces definitions of projective and injective tensor products of convex bodies containing the origin in their interior, using the Hilbert tensor product of Euclidean spaces to define polarity. The projective tensor product is defined as the convex hull of simple tensors, and the injective tensor product as the polar of the projective tensor product of polars. Key properties are established, including a universal property for the projective tensor product, continuity with respect to the Banach-Mazur distance, and duality between projective and injective products. The talk also explores relationships with ellipsoids, particularly the Löwner and John ellipsoids, and introduces the concept of dual tensor products. A Minkowski-type theorem is presented, establishing a bijection between tensor products of convex bodies and tensor norms on finite-dimensional spaces, preserving duality and injectivity/projectivity properties. The talk concludes with an example illustrating the smallest injective tensor product and its relation to sections of infinite-dimensional convex sets.