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Intrinsic volumes of sublevel sets

Benoît Jubin

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{Γ(k/2)}{2 π^{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--Poincaré 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.

Intrinsic volumes of sublevel sets

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 and 0 is a regular value of , then the intrinsic volume of degree of the sublevel set , if the latter is compact, is given by \begin{equation*} \mathcal{L}_{n-k}(M^0) = \frac{Γ(k/2)}{2 π^{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 , where the 's are polynomials given in the text. This includes as special cases the Euler--Poincaré 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.

Paper Structure

This paper contains 10 sections, 12 theorems, 58 equations.

Table of Contents

  1. Main results
  2. Conventions and notation
  3. Intrinsic volumes
  4. Intrinsic volumes of sublevel sets
  5. Sublevel sets and level sets
  6. Intrinsic volumes of sublevel sets
  7. Nodal volumes
  8. Continuity of the intrinsic volumes of sublevel sets
  9. Review of function space topologies
  10. Continuity of the intrinsic volumes

Key Result

Proposition \oldthetheorem

Let $M$ be a manifold, let $p \in \overline{\mathbb{N}_{\geq 1}}$, and let $(f, a) \in \mathop{\mathrm{Reg}}\nolimits^p(M, \mathbb{R})$. Then, $M^a_f$ is a full-dimensional $\mathcal{C}^p$-submanifold with boundary of $M$. Its manifold boundary, equal to its topological boundary, is the $\mathcal{C}

Theorems & Definitions (40)

  • Remark \oldthetheorem
  • Proposition \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • Definition \oldthetheorem
  • Remark \oldthetheorem
  • Theorem \oldthetheorem: Divergence theorem
  • proof
  • Theorem \oldthetheorem
  • ...and 30 more