  We develop a calculus based on zonoids - a special class of convex bodies -
for the expectation of functionals related to a random submanifold $Z$ defined
as the zero set of a smooth vector valued random field on a Riemannian
manifold. We identify a convenient set of hypotheses on the random field under
which we define its zonoid section, an assignment of a zonoid $\zeta(p)$ in the
exterior algebra of the cotangent space at each point $p$ of the manifold. We
prove that the first intrinsic volume of $\zeta(p)$ is the Kac-Rice density of
the expected volume of $Z$, while its center computes the expected current of
integration over $Z$. We show that the intersection of random submanifolds
corresponds to the wedge product of the zonoid sections and that the preimage
corresponds to the pull-back.
  Combining this with the recently developed zonoid algebra, it allows to give
a multiplication structure to the Kac-Rice formulas, resembling that of the
cohomology ring of a manifold. Moreover, it establishes a connection with the
theory of convex bodies and valuations, which includes very deep and difficult
results such as the Alexandrov-Fenchel inequality and the Brunn-Minkowsky
inequality. We export them to this context to prove two analogous new
inequalities for random submanifolds. Applying our results in the context of
Finsler geometry, we prove some new Crofton formulas for the length of curves
and the Holmes-Thompson volumes of submanifolds in a Finsler manifold.
