Tensor Products of Convex Cones, Part II: Closed Cones in Finite-Dimensional Spaces

In part I, we studied tensor products of convex cones in dual pairs of real vector spaces. This paper complements the results of the previous paper with an overview of the most important additional properties in the finite-dimensional case. (i) We show that the projective cone can be identified with the cone of positive linear operators that factor through a simplex cone. (ii) We prove that the projective tensor product of two closed convex cones is once again closed (Tam already proved this for proper cones). (iii) We study the tensor product of a cone with its dual, leading to another proof (and slight extension) of a theorem of Barker and Loewy. (iv) We provide a large class of examples where the projective and injective cones differ. As this paper was being written, this last result was superseded by a result of Aubrun, Lami, Palazuelos and Plavala, who independently showed that the projective cone $E_+ \mathbin{\otimes_\pi} F_+$ is strictly contained in the injective cone $E_+ \mathbin{\otimes_\varepsilon} F_+$ whenever $E_+$ and $F_+$ are closed, proper and generating, with neither $E_+$ nor $F_+$ a simplex cone. Compared to their result, this paper only proves a few special cases.