The Zonoid Algebra, Generalized Mixed Volumes, and Random Determinants

We show that every multilinear map between Euclidean spaces induces a unique, continuous, Minkowski multilinear map of the corresponding real cones of zonoids. Applied to the wedge product of the exterior algebra Λ(V ) of a Euclidean space V, this defines the structure of a commutative, associative, and ordered ring on the space A(V ) of virtual zonoids of Λ(V ), which we call the zonoid algebra of V. Our construction incorporates the notion of mixed volume from convex geometry via the length of zonoids (first intrinsic volume), which can be thought of as their average diameter. We also analyze a similar construction based on the complex wedge product on Cn, which naturally leads to the new notion mixed J-volume. These constructions allow to express the expected absolute determinant of random matrices in terms of zonoids.

This work prepares the ground for a probabilistic intersection theory for compact homogenous spaces. Ongoing work with Paul Breiding, Antonio Lerario and Leo Mathis.