Expectation of a random submanifold: the zonoid section

Mathis, Léo; Stecconi, Michele

Annales Henri Lebesgue, Tome 7 (2024), pp. 903-967

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 $\zeta \left(p\right)$ in the exterior algebra of the cotangent space at each point $p$ of the manifold. We prove that the first intrinsic volume of $\zeta \left(p\right)$ 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 deep results such as the Alexandrov–Fenchel inequality and the Brunn–Minkowski 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.

Nous développons un calcul basé sur les zonoïdes – une classe particulière de corps convexes – pour l’espérance de fonctionnelles liées à une sous variété aléatoire $Z$ définie comme l’ensemble des zéros d’un champ aléatoire lisse à valeurs vectorielles dans une variété riemannienne. Nous identifions un ensemble d’hypothèses pour le champ aléatoire sous lesquelles nous pouvons définir sa section en zonoïdes, l’attribution d’un zonoïde $\zeta \left(p\right)$ dans l’algèbre externe de l’espace cotangent à chaque point $p$ de la variété. Nous démontrons que le premier volume intrinsèque de $\zeta \left(p\right)$ est la densité de Kac–Rice du volume moyen de $Z$, tandis que son centre correspond au courant moyen d’intégration sur $Z$. Nous prouvons que l’intersection de sous variétés indépendantes correspond au produit extérieur des sections en zonoïdes et que la préimage correspond au pull back.

La combinaison de ces résultats avec l’algèbre des zonoïdes récemment développée, permet de donner une structure multiplicative aux formules de Kac–Rice qui évoque celle d’un anneau de cohomologie d’une variété. En outre, cela permet d’établir une connection avec la théorie des corps convexes et des valuations, qui contiend des résultats profonds tels que l’inégalité d’Alexandrov–Fenchel ou de Brunn–Minkowski. Nous exportons ces résultats dans notre contexte pour produire deux nouvelles inégalités analogues pour les sous variétés aléatoires. En appliquant nos résultats dans le contexte de la géométrie Finsler, nous prouvons des nouvelles formules de Crofton bour la longueurs de courbes et le volume de Holmes–Thompson des sous variétés d’une variété finslerienne.

Keywords: Kac–Rice Formula, zonoids, random fields, convex bodies

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