The Zonoid Algebra, Mixed Volumes and Random Determinants (Part 1)
- Paul Breiding (MPI MiS, Leipzig)
I will introduce the zonoid algebra. Starting from the monoid structure of zonoids in a d-dimensional real vector space I will explain how to turn this structure into an algebra, where we can “multiply” zonoids. More specifically, I will show that every multilinear map between finite dimensional vector spaces has a unique, continuous, Minkowski multilinear extension to the corresponding space of zonoids. Taking the wedge product of vector spaces as the multilinear map, we get a definition of the wedge of zonoids. This is the definition of the product in our algebra. The motivation for this construction comes from probabilistic intersection theory in a compact homogeneous space, where the zonoid algebra plays the role of a probabilistic cohomology ring.