In this work, we present a new method of computation that we call zonoid calculus. It is based
on a particular class of convex bodies called zonoids and on a representation of zonoids using
random vectors. Concretely, this is a recipe to build multilinear maps on spaces of zonoids from
multilinear maps on the underlying vector spaces. We call this recipe the fundamental theorem
of zonoid calculus (FTZC).
Using this and the wedge product in the exterior algebra we build the zonoid algebra, that is a
structure of algebra on the space of convex bodies of the exterior algebra of a vector space. We
show how this relates to the notion of mixed volume on one side and to random determinants
on the other. This produces new inequalities on expected absolute determinants. We also show
how this applies in two detailed examples: fiber bodies and non centered Gaussian determinants.
We then use FTZC to produce a new function on zonoids of a complex vector space that we call
the mixed J-volume.
We uncover a link between the zonoid algebra and the algebra of valuations on convex bodies.
We prove that the wedge product of zonoids extends Alesker’s product of smooth valuations.
Finally we apply the previous results to integral geometry in two different context. First we
show how, in Riemannian homogeneous spaces, the expected volume of random intersections
can be computed in the zonoid algebra. We use this to produce a new inequality modelled
on the Alexandrov–Fenchel inequality, and to compute formulas for random intersection of real
submanifolds in complex projective space. Secondly, we prove how a Kac-Rice type formula can
relate to the zonoid algebra and a certain zonoid section. We use this to study the expected
volume of random submanifolds given as the zero set of a random function. We again produce an
inequality on the densities of expected volume modelled on the Alexandrov–Fenchel inequality,
as well as a general Crofton formula in Finsler geometry.
