Intrinsic volumes of sublevel sets

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.


Introduction
Intrinsic volumes are geometric invariants attached to well-behaved subsets of Riemannian manifolds.They include the volume and the Euler-Poincaré characteristic.Among their applications in the field of integral geometry are Weyl's tube formula ( [21]), that gives the volumes of tubular neighborhoods of submanifolds, and the kinematic formula of Blaschke, Chern and Santaló ( [8,9]), that gives the volume of the Minkowski sum of two convex bodies.They were introduced in their modern form by Herbert Federer in the seminal article [10], where they are called curvature measures, after special cases in convex geometry were treated by Hermann Minkowski.Among the vast literature on their subject, we only mention the book [17], the survey on a related topic [20], and the articles [12,13,23].
In this article, we study the intrinsic volumes of sublevel sets of functions defined on Riemannian manifolds.These were already studied from the point of view of Morse theory in [11].Since intrinsic volumes include the volume of the boundary, this study encompasses volumes of level sets, and in particular of zero sets, also called nodal sets.The first closed explicit formulas computing nodal volumes appeared in [3], which was a motivation for the present article.These formulas can be seen as generalizations of the so-called Kac-Rice formula (see for instance [18]).
Sublevel sets are also studied in probability theory, where superlevel sets of random fields are called excursion sets; see for instance the books [1,2] and the articles [3,16,22].The importance of the formulas obtained in this paper for the study of random fields (as studied in [3]), compared to existing Kac-Rice formulas, stems from the fact that they are in "closed form" as opposed to being limits of integrals depending on a parameter.

Main results
We now describe the contents of this article in more detail.In this introduction, we restrict ourselves to the flat case.In Section 1, we recall the definition and main properties of intrinsic volumes.If N is a flat compact n-dimensional Riemannian manifold with boundary, they take the form for 0 ≤ k ≤ n, where b k ∈ R and S is the second fundamental form of ∂N in N .
In Section 2, we specialize our study to the case where N is a sublevel set.Namely, let M be a flat n-dimensional Riemannian manifold (without boundary), let f ∈ C 3 (M, R), and assume that a ∈ R is a regular value of f and that the sublevel set M a := f −1 (]−∞, a]) is compact.The second fundamental form of ∂M a in M can be expressed in terms of the gradient and the Hessian of f .An important lemma (Lemma 2.2) establishes that the above integrand is then a polynomial in ∇ f and Hess(f ) divided by ∇ f 2(k−1) .We then use the divergence theorem to transform the above integral over ∂M a into an integral over M a .This leads to our main formula which, in the flat case, reads for 1 ≤ k ≤ n, where the P n,k 's are polynomials given in the text (Theorem 2.9).The main advantage of this formula is that it is an explicit integral over M a (and not ∂M a ) of a continuous functional in f and its derivatives up to order 3. Since the intrinsic volume of degree n − 1 is half the volume of the boundary, this formula can be used to compute the volume of level sets.If the ambient manifold M is compact, one can use the intrinsic volume of either the sublevel or the superlevel set, yielding for the volume of the zero set Z f of f the formula where σ f is the sign of f and η f := f 2 + ∇ f 2 .This formula was obtained in the case of a flat torus in [3].As in [3], one can do an integration by parts to eliminate σ f and obtain an integral over M where the integrand is a Lipschitz continuous functional of f and its derivatives up to order 2 (Equation ( 49)).This regularity allows one to apply techniques of the Malliavin calculus to obtain results about the expected value, variance and higher moments of the nodal volumes of certain families of random fields (see [3]), and more generally of the intrinsic volumes of their excursion sets.In Section 3, after recalling basic facts on natural topologies on C p (M, R), the uniform and the (Whitney) strong C p -topologies, we prove that conditions needed to establish our formulas (regularity of the value and compactness of the sublevel sets) are generic.Then, we prove the continuity of intrinsic volumes of sublevel sets.For instance, if 0 ≤ k ≤ n, then the function is Lipschitz continuous, where the domain is the set of couples (f, a) where f ∈ C 3 (M, R) is proper bounded below and a ∈ R is a regular value of f , and is equipped with the uniform C 3 -topology (Theorem 3.10).In particular, the Euler-Poincaré characteristic of sublevel sets is locally constant.

Conventions and notation
• If P is a proposition, then [P ] := 1 if P else 0.
• We denote by pr i the projection on the i th factor of a direct product.
• The bracket − : R → Z denotes the floor function.
• The symbol (resp.) denotes the symmetric (resp.exterior) product or power of vector spaces.
• Unless otherwise specified, manifolds are Hausdorff, paracompact, real, finite-dimensional, and smooth, that is, of class C ∞ .
• The space of smooth sections of the vector bundle E → M is denoted by Γ(E → M ) or Γ (p) (E → M ) if the differentiability class p ∈ N need be specified.For instance, the metric tensor of a Riemannian manifold M is an element of Γ( 2 T * M → M ).

Intrinsic volumes
Let (M, g) be an n-dimensional compact Riemannian manifold with boundary.Its metric will also be denoted by −, − and the associated norm by − .We denote by ∇ lc its Levi-Civita connection.Let vol M be the Riemannian density on M and vol ∂M be the induced density on ∂M .The symbol vol will also denote the associated volume of (sub)manifolds.Let R ∈ Γ( 2 2 T * M → M ) be the covariant curvature tensor of M .Let S := ((∇ lc ν| T ∂M ) T ) ∈ Γ( 2 T * ∂M → ∂M ) be the second fundamental form of ∂M in M , where ν ∈ Γ(T M | ∂M → ∂M ) is the outward unit normal vectorfield on ∂M and (−) T : T M | ∂M → T ∂M denotes the tangential component, and : T ∂M → T * ∂M denotes the musical isomorphism induced by the metric.The symbol tr denotes the trace of a bilinear form.
For the exterior product of symmetric bilinear forms, ∧ : 2 p V × 2 q V → 2 p+q V, also called in differential geometry the Kulkarni-Nomizu product, see for instance [10, §2].
For 0 ≤ k ≤ n, the intrinsic volume of degree n − k of M is defined as where We also set One has The first two equalities immediately follow from a 0 = 1 and from a 1 = 0 and b 1 = 1 2 respectively (indeed, tr( 0 R x ) is the trace of the identity of 0 T * x M R, which is 1, and similarly for tr( 0 S x )).The third equality is the Gauss-Bonnet-Chern theorem (see [6,7] for the original articles, and [19] for manifolds with boundary), where the right-hand side is the Euler-Poincaré characteristic of M , and in particular is an integer and is zero in the odd-dimensional boundaryless case. Since where scal denotes the scalar curvature of M .Note that tr S is (n − 1) times the mean curvature of ∂M in M .If M is flat, that is, R = 0, then Formula (5) simplifies, since only the summand corresponding to m = 0 may be nonzero, giving for 1 ≤ k ≤ n.In particular, Note that tr S is (n − 1) times the mean curvature, and det S "the" Lipschitz-Killing, or Gauss-Kronecker, curvature, of ∂M in M .
2 Intrinsic volumes of sublevel sets

Sublevel sets and level sets
If f : M → R is a function on a set and a ∈ R, then the a-sublevel set of f is defined by also written M a if there is no risk of confusion, and the a-level set of f is f −1 (a).
Let M be a manifold, let p ∈ N ≥1 , and let f ∈ C p (M, R).The real number a ∈ R is a regular value of f if f (x) = a implies d f (x) = 0 for all x ∈ M .We define the sets We also set Proof.By the submersion theorem, f −1 (a) is a C p -hypersurface of M .By considering separately points x ∈ M such that f (x) < a and such that f (x) = a, one checks that M a f is a full-dimensional C p -submanifold with boundary of M , and that ∂M a f = f −1 (a).

Intrinsic volumes of sublevel sets
Let (M, g) be an n-dimensional Riemannian manifold (not necessarily compact, but without boundary).Let f ∈ C 2 (M, R).Its gradient is defined by ∇ f := (d f ) .Its Hessian is defined by Hess(f ) := ∇ lc d f = (∇ lc ∇ f ) .Its Laplacian is the trace of its Hessian, ∆ f := tr(Hess(f )).Let (f, a) ∈ Reg 2 (M, R).By Proposition 2.1, the set M a is a full-dimensional C 2submanifold with boundary of M and its boundary is the We briefly explain the idea underlying the rest of this subsection.By Formula ( 19), the integrals in the sum on m in Formula ( 5) are equal to We will convert these integrals on ∂M a into integrals on M a by using the divergence theorem.To do this, we need to find a vectorfield X ∈ X(M a ) such that Besides the boundary, the possibly problematic points are the points where ∇ f = 0, first because of the factor ∇ f 2m+1−k , and also because of the restriction to ∇ f ⊥ .Since the two regions of interest are at f = a and at ∇ f = 0, it makes sense to look for a vectorfield of the form and F vanishes sufficiently fast at 0 for the divergence to be integrable.We now make this idea precise.
2 , there exists a homogeneous polynomial P n,k,m with integer coefficients such that for any ndimensional Euclidean space with orthonormal basis (V, B), any symmetric bilinear forms R ∈ 2 2 V * and H ∈ 2 V * , and any v ∈ V \ {0}, one has where (r ijkl ) (resp.(h ij ) and For the sake of definiteness, if B = (e i ) 1≤i≤n is an orthonormal basis of V, we consider the basis (e i ∧ e j ) 1≤i<j≤n of 2 V.The coefficients of R can be written (r ijkl ) 1≤i,j,k,l≤n with i < j and k < l and (i, j) ≤ (k, l) in the lexicographic order.
Proof.Let n, k, m, (V, B), R, H, v be as in the statement.Without loss of generality, we can suppose that (V, B) = (R n , std) with the standard inner product.Set In particular, a 1 = v 1 and a n = v .We first assume that v 1 > 0. Let P be the following change of basis matrix: It is an orthogonal matrix and P −1 HP restricted to the rows and columns 2 ≤ i, j ≤ n is the matrix of H| v ⊥ in an orthonormal basis.
Since the β ij 's are polynomials in the v k 's and a k 's, the coefficients (P −1 HP ) ij with 2 ≤ i, j ≤ n are of the form The change of basis matrix in 2 V associated with P , say Q, has coefficients Q ijkl = P ik P jl − P il P jk .As with P , the matrix Q −1 RQ restricted to the rows and columns 2 ≤ i, j, k, l ≤ n (with i < j and k < l) is the matrix of R| v ⊥ in an orthonormal basis.The coefficients (Q −1 RQ) ijkl with 2 ≤ i, j, k, l ≤ n (with i < j and k < l) are of the form The coefficients of the exterior product of their exterior powers is again of a similar form, hence so is its trace.More precisely, it is a rational fraction with variables r ijkl , h ij , v i , a i .The denominator is a product of a i 's, where the exponent of a n is at most This expression was obtained under the assumption that v 1 > 0, but it is intrinsic to (R, S, v) and invariant under orthogonal transformations of V. Therefore, it also holds if v = e n , in which case all the a i 's with i < n vanish.As a consequence, the only a i 's at the denominator are those with i = n, that is, a n = v .
The variable a n does not appear in the numerator (since β ij only involves a j−1 ).For i < n, then a i as a function of the v k 's is not differentiable at e n but is differentiable at e 1 , so by invariance under orthogonal transformation, the variables a i with i < n can only appear in the numerator with even exponents.Therefore, the numerator is a polynomial in the coefficients r ijkl , h ij , v i .
2 , we define P n,k,m to be the (unique) polynomial whose existence is asserted in Lemma 2.2.For other values of the indices, we set P n,k,m := 0. We set P n,k := P n,k,0 .The proof shows how to compute the P n,k,m 's.For instance, one has for 2 ≤ i, j ≤ n, and the trace of an exterior power can be computed as a sum of minors of given order.We consider a few special cases: , then the left-hand side of ( 21) is the trace of the identity on 0 v ⊥ R, so P n,1 = 1.
• For the case k = 2 (hence m = 0), note that tr( ).This gives The first such polynomial is This is simply H(u, u) where u is any of the two unit vectors orthogonal to v. .
A double expansion of this determinant yields In view of the special cases considered above, and under the nondegeneracy condition for the third equation, one has We now state the version of the divergence theorem that will be useful to us.The divergence of a C 1 -vectorfield X ∈ X(M ) is defined by L X vol M = (div X) vol M , where L denotes the Lie derivative.Many generalizations of the standard divergence theorem have been proved, relaxing hypotheses on the regularity and compactness of the manifold or stratified space and on the regularity of the vectorfield, encompassing the present statement.We include a proof for the convenience of the reader.Theorem 2.5 (Divergence theorem).Let (M, g) be a compact n-dimensional Riemannian manifold with boundary.Let X ∈ X(M ) be a continuous vectorfield on M which is of class Proof.If X is of class C 1 on M , then this is the standard divergence theorem.Else, we consider the geodesic flow from the boundary of M along the outward unit normal vectorfield ν.For > 0 small enough, set θ : ∂M → M, x → exp(x, − ν x ) and set M := M \ s∈[0, [ θ s (∂M ).For small enough, M is a compact submanifold with boundary of M , and θ induces a diffeomorphism ϑ : ∂M ∼ − → ∂M .Applying the standard divergence theorem on M , one obtains M (div X) vol M = ∂M X, v vol ∂M .When → 0, the left-hand side converges to M (div X) vol M by Lebesgue's dominated convergence theorem, since div X ∈ L 1 (M ).The right-hand side is equal, by change of variable, to ∂M ϑ * X, v (det T ϑ ) vol ∂M , which converges to ∂M X, v vol ∂M since the integrand is uniformly convergent and ∂M is compact.
We can now prove a first general result.
under the condition that the divergence appearing in the integral exists and is integrable.
under the same conditions.
Remark 2.7.By "F (u) ∼ ∞ u u ", we mean that lim u →+∞ d F (u), u u = 0, where d is the distance on T M induced by the Riemannian metric of M (or any distance, since M a is compact and u u has unit norm).
Proof.Starting with the definition (5), we use the expression of the second fundamental form (19) and Equation ( 23) to obtain The asymptotic property of F k,m ensures that the vectorfield whose divergence is considered in the statement is continuous on ∂M a and its value there is Finally, the hypotheses of the proposition ensure that the divergence theorem applies.
Remark 2.8.Since P n,1 = 1, the theorem for k = 1 holds for (f, a) ∈ Reg 2 c (M, R).Our next step is to find explicit functions F (in particular proving that some exist) making the divergence appearing in the theorem integrable.We consider radial maps of the form . We set for ≥ 0. These choices for F yield the following theorem.
Theorem 2.9.Let (M, g) be an n-dimensional Riemannian manifold.Let (f, a) ∈ Reg 3 c (M, R).For 0 ≤ k ≤ n, one has If M is flat and 1 ≤ k ≤ n, then For k = 1, 2, this gives Similarly, when M is flat and Hess Remark 2.10.There are obviously many natural choices for the functions F and G.For instance, one can take F k,m := F k .With the F k 's given above, the divergence corresponding to the m th summand reads div ∇ f 2m P n,k,m (R,Hess(f ),∇ f ) ∇ f .In the case of nodal volumes, other choices are given in the next subsection.
Remark 2.11.Intrinsic volumes can be defined for Riemannian manifolds with corners, and even Whitney stratified spaces of "positive reach" in manifolds.Since the divergence theorem admits generalizations to these settings, it is possible to extend the above results to sublevel sets of functions defined on Riemannian manifolds with boundary or corners, and to Whitney stratified spaces in Riemannian manifolds, under the assumption that the function is transverse to the boundary or the strata respectively.Boundary terms will appear in the formulas.We do not carry out this generalization in full and only give a formula for nodal volumes in the next subsection (see Remark 2.13).

Nodal volumes
In this subsection, we show how we can compute the intrinsic volumes of the zero sets, or nodal sets, of functions defined on compact Riemannian manifolds.Let (M, g) be a 34) gives an integral on M .Using the general formula of Theorem 2.6 yields (where minus the absolute value appears since f is negative on M 0 f and positive on M 0 −f ).Of course, this identity could have been obtained directly by applying the divergence theorem to the identity vol Recalling the definition of η f, by Equation (31), we set We also write σ f : M → {−1, 0, 1} for the sign of f .Setting, in Formula (37), (see [15] for the computation details).
Remark 2.12.In the last three formulas, all terms of the integrands are bounded on M and continuous on M \ Z f .Indeed, the Hessian expressions are quadratic in ∇ f , the arctan and tanh expressions are linear in ∇ f when ∇ f is small, and the cosh expression is exponentially small in |f | when |f | is small.However, not all terms need be continuous on M .This problem is dealt with below.
Remark 2.13.Fulfilling the promise made in Remark 2.11, let M be a compact Riemannian manifold with boundary.If f intersects ∂M transversely, then Formula (37) becomes Note that by the transversality assumption, Z f ∩ ∂M is negligible in ∂M .This formula reduces in dimension 1 to [3,Prop. 3].
More generally, the intrinsic volumes of subsets have the additivity property when A, B are subsets of a compact n-dimensional Riemannian manifold M such that all terms are well-defined (see [10,Thm. 5.16(6)]).Therefore, if f ∈ C 3 0−reg (M, R), then The terms corresponding to the first summand in (32) cancel out, so that The exponent k of σ f is congruent modulo 2 to deg Hess(f In particular, L n−k (Z f ) = 0 for k even, as expected.One can also consider Z f as a Riemmannian manifold with curvature R and obtain where R is given by the Gauss formula for the curvature of submanifolds, R(X, We return to the question raised in Remark 2.12 of having continuous integrands.The only non-continuous terms in the integrands of Equations ( 39), ( 40), (41) are of the form . This is dealt with in [3] (in the case of Equation (39) on a flat torus) using an integration by parts.The same method extends to compact Riemannian manifolds as follows.One has Therefore, We temporarily assume that f is of class C 3 and we use the fact that div ) has a vanishing integral on M (by the standard divergence theorem).Therefore, where the norm of the Hessian is the Hilbert-Schmidt norm.Therefore, the third derivatives cancel out.Since C 2 (M, R) is dense in C 3 (M, R) for the (Whitney) strong C 2 -topology (see for instance [14, Thm.II.2.6], and the next section for function space topologies) and the involved quantities are continuous in this topology, one has, for any For example, one has In the case of a flat torus, this is [3,Prop. 7].
Remark 2.15.These formulas can also be written in terms of the tracefree Hessian.Recall that Hess 0 (f ) = Hess(f ) − ∆ f n id.A tracefree linear map is Hilbert-Schmidt-orthogonal to the identity, so Hess In Equation (49), the integrand is a Lipschitz continuous functional of f ∈ C 2 0−reg (M, R) (see next section for the precise setting), so one can apply techniques of the Malliavin calculus (see [3]).The only difference between (49) and [3,Prop. 7] is the additional term involving the Ricci curvature, |f |η −3  f Ric(∇ f, ∇ f ), and this term is in the required domain of the Malliavin calculus by the same proof as [3, Lem. 2 p. 26].Therefore, [3, Thm.1] holds on any compact Riemannian manifold.Similarly, Formula (42) shows that the extra boundary terms are not problematic, so [3, Thm.1] holds on any compact Riemannian manifold with corners, a generalization which includes [3, Thm.2] as a special case.

Continuity of the intrinsic volumes of sublevel sets 3.1 Review of function space topologies
For this subsection, we refer to [14, Ch.II] for details.Let (M, g) be a Riemannian manifold and p ∈ N. We will use two different topologies on the set C p (M, R), the uniform C p -topology, and the finer (Whitney) strong C p -topology.The resulting topological spaces will be denoted with the subscripts U and S respectively.The first is a completely metrizable group and the second is a Baire topological group (countable intersections of dense open subsets are dense).In particular, there is a notion of Lipschitz continuity (by which we mean "local Lipschitz continuity"1 ) for maps between the first space and other metric spaces.The product C p (M, R) × R will be considered with the corresponding product topology, and the sets Reg p (M, R) and Reg p c (M, R) defined in Equations ( 17) and ( 18) with the corresponding subspace topologies. If We denote by ∇ lc i f (x) the norm of this multilinear form induced by the norm g x on T * x M , and by ∇ lc i f ∞ the supremum of these norms for x ∈ M .Let p ∈ N.For the uniform C p -topology, a neighborhood basis of 0 is given by for ∈ R >0 .For the strong C p -topology, a neighborhood basis of 0 is given by for ∈ C 0 (M, R >0 ).The uniform and strong C ∞ -topologies are obtained as the unions of the corresponding C p -topologies.The strong C p -topology does not depend on the Riemannian metric (it could actually be defined using norms of usual derivatives in charts).These topologies differ in the control of functions at infinity (in particular, they are equal when M is compact).Results involving the strong topology will often remain true for the uniform topology when restricted to proper functions.
The strong topology has the disadvantage that the inclusion of constant functions, R → C p (M, R) S , a → (x → a), is not continuous.For example, the function is Lipschitz continuous, but the analogous result (for mere continuity) with the strong topology does not hold.Therefore, when studying sublevel sets at varying heights, we will use the uniform topology and we will restrict our attention to proper functions, and when studying sublevel sets at a fixed height, we will use the strong topology if we want to allow nonproper functions.We denote by C p p (M, R) (resp.Example 3.2.For a given function, the set of real numbers such that the associated sublevel set is compact can be any downset.Indeed, consider the functions on R which send x to respectively x or a or a + e x or x 2 .The sets of real numbers such that the associated sublevel set is compact are ∅ and ]−∞, a[ and ]−∞, a] and R respectively.If in the second case one requires that a be a regular value, then consider x → a − e x . Recall that η f was defined by Equation (38) and τ by Equation (52).

Lemma 3.3. The function
As for density, by the Morse-Sard theorem, if f ∈ C n (M, R), then the set of regular values of f is dense.This implies that Reg [14,Thm. II.2.6]).
For C p 0−reg (M, R), one can use the transversality theorem as follows.Let f ∈ C p (M, R) and ∈ C 1 (M, R >0 ).Let (φ i ) i∈I be a smooth partition of unity subordinated to some locally finite atlas of M .Consider the map Φ : . Then, Φ is submersive, so for almost all tuples (λ i ), the map Φ(−, (λ i )) is transverse to 0.
The proofs work similarly with the compactness requirement added.

Continuity of the intrinsic volumes
We begin with the special cases of the volume and the nodal volume, which will be needed in the proof of the general case.We actually prove a more general statement, where denotes the symmetric difference of two sets.
Proposition 3.5.Let (M, g) be a Riemannian manifold.The functions are continuous (resp.Lipschitz continuous) when the domains are given the uniform C 0 (resp.C 1 )-topology.The functions are continuous when the domains are given the strong C 0 -topology.The function is continuous when the domain is given the uniform C 1 -topology.
Remark 3.6.The continuity of nodal volumes was proved in the Euclidean case in [4, Thm.3], with a similar proof.
We first prove a lemma.
Lemma 3.7.Let f ∈ C 1 0−reg,c (M, R).For any neighborhood U of Z f , there exists an open neighborhood V of f in the strong C 0 -topology (and, if f is proper, in the uniform C 0 -topology) such that for any h, k ∈ V , one has Z h ⊆ U and M 0 h M 0 k ⊆ U .
Proof.Let f and U be as in the statement.Then In the proper case, one has 1 := inf{f (x) | x ∈ M \ K} > 0 and we set V := f + U 0 (min( 0 , 1 )).In the nonproper case, let 2 := min( 0 , inf{f (x) | x ∈ ∂K}) > 0, let ∈ C 0 (M, R >0 ) be the function equal to 2 on K and min( 2 , f ) on M \ K, and set Proof of the proposition.(i) Let Φ be the first function in the proposition.Since M a f = M 0 τ (f,a) and τ is Lipschitz continuous, it suffices to consider 0-sublevel sets.Let ((f, 0), (g, 0 ))+Φ((g, 0), (k, 0)).Therefore, it suffices to prove the Lipschitz continuity of Φ on the diagonal.Continuity is proved by Lemma 3.7 and we defer the proof of Lipschitz continuity to the end of the proof.
(ii) In the non-proper case, it is similarly sufficient to prove the continuity of the third function, say Ψ, on the diagonal, and the Lemma 3.7 also proves the result.
(iv) We now prove continuity of nodal volumes.By Lipschitz continuity of τ , we can restrict our attention to (f, 0) ∈ Reg 1 pb (M, R) U .Then, Z f is a compact C 1 -hypersurface.For any x ∈ Z f , there exists a smooth chart φ : Since Z f is compact, there exists a finite cover (U i ) i∈I of Z f by such sets.Let (ψ i ) i∈I be a smooth partition of unity subordinated to (U i ) i∈I .
By Lemma 3.7, there exists a neighborhood V of f such that the nodal set of any h ∈ V is included in i∈I U i .For any i ∈ I, by the implicit function theorem, there exists a neighborhood V i ⊆ V of f in the C 1 -topology such that for any h ∈ V i , the hypersurface where pr d−1 : R d → R d−1 is the projection on the first d − 1 components.On this expression, the continuity of vol(Z − ) for the uniform C 1 -topology is clear.(v) We now prove the Lipschitz continuity statement.Let (f, 0) ∈ Reg 1 pb (M, R).By Lemma 3.7, there exists an open neighborhood V of f in the uniform C 0 -topology such that Let K be a compact neighborhood of Z f such that := Since by the previous part of the proof, the nodal volume is continuous, vol(Z h ) is bounded, say by A > 0, on a C 1 -neighborhood W of f .Therefore, if h, k ∈ C 1 0−reg,c (M, R)∩ (f + U 1 ( )) ∩ W , then vol(M 0 h M 0 k ) ≤ A h − k ∞ where A > 0 only depends on A and (M, g).
Remark 3.8.The volume function is not uniformly continuous, as the pairs of constant functions equal to ± 1 n on a nonempty compact manifold M show: the volume of the 0sublevel set jumps from 0 to vol(M ) for two arbitrarily close functions.Similar examples can be given for any nonzero intrinsic volume.
Remark 3.9.The proof shows that the first two functions are actually pointwise Lipschitz continuous when the domain is given the uniform C 0 -topology.

8
variables.By homogeneity considerations, P n,k,m has degree 2(k − 1) in the coefficients of v, degree k − 1 − 2m in the coefficients of H, and degree m in the coefficients of R.

Proposition 3 . 1 .
C p b (M, R)) the set of proper (resp.bounded below) functions in C p (M, R), and by C p pb (M, R) := C p p (M, R)∩C p b (M, R) the set of proper bounded below functions.Similarly, we set Reg p * := Reg p (M, R) ∩ (C p * (M, R) × R) for * = b, p, pb.Let p ∈ N. At least one (resp.all) sublevel set(s) of f ∈ C p (M, R) is/are compact if and only if f is bounded below (resp.proper bounded below).In particular, Reg p pb (M, R) ⊆ Reg p c (M, R).The subsets C p b (M, R) and C p p (M, R) and C p pb (M, R) are open and closed in C p (M, R) U .Proof.Obvious.

Proof. Obvious. Proposition 3 . 4 .
Let p ∈ N ≥1 .The subset Reg p p (M, R) is open and dense in C p p (M, R) U × R. The subset C p 0−reg (M, R) (resp.C p 0−reg,c (M, R)) is open and dense in C p (M, R) S (resp.C p 0−c (M, R) S ).The three openness results actually hold for the (uniform or strong) C 0topology.Proof.One has, Reg p p

the Lipschitz-Killing curvatures of
∂M in M .More general versions L n−k (M, A) can be defined for Borel subsets A ⊆ M and are called curvature measures in M .The intrinsic volumes are the total measures of these curvature measures, that is,