The zonoid algebra, generalized mixed volumes, and random determinants

TL;DR

This paper introduces the zonoid algebra, a new algebraic structure on zonoids derived from multilinear maps, which generalizes mixed volumes and connects to random determinants, offering fresh insights into convex geometry.

Contribution

It constructs a novel algebraic framework on zonoids using multilinear maps, generalizing mixed volumes and introducing the concept of mixed J-volume.

Findings

Defines the zonoid algebra as a commutative, associative, and partially ordered ring.

Introduces new functionals on zonoids, generalizing mixed volume.

Connects the framework to the theory of random determinants.

Abstract

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 of a Euclidean space, this yields a multiplication of zonoids, defining the structure of a commutative, associative, and partially ordered ring, which we call the zonoid algebra. This framework gives a new perspective on classical objects in convex geometry, and it allows to introduce new functionals on zonoids, in particular generalizing the notion of mixed volume. We also analyze a similar construction based on the complex wedge product, which leads to the new notion of mixed $J$-volume. These ideas connect to the theory of random determinants.