Expectation of a random submanifold: the zonoid section

TL;DR

This paper introduces a zonoid-based calculus for analyzing the expected geometric properties of random submanifolds, establishing new inequalities and formulas that connect convex geometry with stochastic geometry on manifolds.

Contribution

It develops a novel zonoid section framework to compute expectations of submanifold functionals and links convex body theory with random geometry, leading to new inequalities and Crofton formulas.

Findings

Zonoid section's first intrinsic volume equals Kac-Rice density.

Expected current of integration over the submanifold is given by the zonoid center.

New inequalities for random submanifolds analogous to classical convex inequalities.

Abstract

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.