Intrinsic volumes of sublevel sets

TL;DR

This paper derives formulas for intrinsic volumes of sublevel sets of functions on Riemannian manifolds, generalizing classical results like the Euler characteristic and Kac-Rice formula, and proves their Lipschitz continuity.

Contribution

It introduces new integral formulas for intrinsic volumes of sublevel sets on Riemannian manifolds, extending classical geometric and probabilistic results.

Findings

Formulas for intrinsic volumes as integrals involving function derivatives

Special cases include Euler characteristic and nodal volumes

Proves Lipschitz continuity of intrinsic volumes

Abstract

We establish formulas that give the intrinsic volumes, or curvature measures, of sublevel sets of functions defined on Riemannian manifolds as integrals of functionals of the function and its derivatives. For instance, in the Euclidean case, if $f \in \mathcal{C}^3(\mathbb{R}^n, \mathbb{R})$ and 0 is a regular value of $f$, then the intrinsic volume of degree $n-k$ of the sublevel set $M^0 = f^{-1}(]-\infty, 0])$, if the latter is compact, is given by \begin{equation*} \mathcal{L}_{n-k}(M^0) = \frac{\Gamma(k/2)}{2 \pi^{k/2} (k-1)!} \int_{M^0} \operatorname{div} \left( \frac{P_{n, k}(\operatorname{Hess}(f), \nabla f)}{\sqrt{f^{2(3k-2)} + \|\nabla f\|^{2(3k-2)}}} \nabla f \right) \operatorname{vol}_n \end{equation*} for $1 \leq k \leq n$, where the $P_{n, k}$'s are polynomials given in the text. This includes as special cases the Euler--Poincar\'e characteristic of sublevel sets and the nodal volumes of functions defined on Riemannian manifolds. Therefore, these formulas give what can be seen as generalizations of the Kac--Rice formula. Finally, we use these formulas to prove the Lipschitz continuity of the intrinsic volumes of sublevel sets.