Nicolas Dutertre
email: nicolas.dutertre@univ-angers.fr

PRINCIPAL KINEMATIC FORMULAS FOR GERMS OF CLOSED DEFINABLE SETS

Keywords: 2010 Mathematics Subject Classification. 53C65, 14P10, 32B20 Kinematic formulas, definable sets, Lipschitz-Killing curvatures, polar invariants

We prove two principal kinematic formulas for germs of closed definable sets in R n , that generalize the Cauchy-Crofton formula for the density due to Comte and the infinitesimal linear kinematic formula due to the author. In this setting, we do not integrate on the space of euclidian motions SO(n) R n , but on the manifold SO(n) × S n-1 .

Introduction

The search for kinematic formulas is one of the main goal of integral geometry. Such formulas have been proved in various contexts by various authors, for instance:

-For convex bodies by Blaschke and Hadwiger (see [START_REF] Klain | Introduction to geometric probability[END_REF]); -For manifolds by Chern [START_REF] Chern | On the kinematic formula in integral geometry[END_REF] and manifolds with boundary by Santaló [START_REF] Santaló | Integral geometry and geometric probability[END_REF]; -For PL-sets by Cheeger, Müller and Schrader [START_REF] Cheeger | Kinematic and tube formulas for piecewise linear spaces[END_REF]; -For sets with positive reach by Federer [START_REF] Federer | Curvature measures[END_REF][START_REF] Federer | Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften[END_REF] (see also [START_REF] Rother | A short proof of principal kinematic formula and extensions[END_REF]); -For subanalytic sets by Fu [START_REF] Fu | Curvature measures of subanalytic sets[END_REF], and more generally for sets definable in an o-minimal structure by Bernig, Bröcker and Kuppe [START_REF] Bröcker | Integral geometry of tame sets[END_REF][START_REF] Bernig | Lipschitz-Killing invariants[END_REF][START_REF] Bernig | Courbures intrinsèques dans les catégories analyticogéométriques[END_REF]. There are many other situations where kinematic formulas hold, but we cannot give here a complete list of all the interesting papers published on this topics.

In this paper, we are interested in the case of definable sets in an o-minimal structure. Definable sets are a generalization of semi-algebraic sets and globally subanalytic sets, we refer the reader to classical references [START_REF] Van Den Dries | Geometric categories and o-minimal structures[END_REF][START_REF] Van Den Dries | Tame topology and o-minimal structures[END_REF][START_REF] Coste | An introduction to o-minimal geometry[END_REF][START_REF] Loi | Lecture 1: o-minimal structures[END_REF][START_REF] Rolin | A survey on o-minimal structures[END_REF] for basic definitions and results on this topics. The study of the geometric properties of these objects was initiated by Fu [START_REF] Fu | Curvature measures of subanalytic sets[END_REF], who developed integral geometry for compact subanalytic sets. Using the technology of the normal cycle, he associated with every compact subanalytic set X of R n a sequence of curvature measures Λ 0 (X, -), . . . , Λ n (X, -), called the Lipschitz-Killing measures, and he established several integral geometry formulas. Among them, he proved the following kinematic formulas: for k ∈ {0, . . . , n}, we have

SO(n) R n Λ k (X ∩ γY, U ∩ γV )dγdx = p+q=k+n e(p, q, n)Λ p (X, U )Λ q (X, V ),
where X and Y are two compact subanalytic subsets of R n and U and V are Borel subsets of X and Y respectively. We will state these formulas specifically in the next section. In [START_REF] Bröcker | Integral geometry of tame sets[END_REF] (see also [START_REF] Bernig | Lipschitz-Killing invariants[END_REF][START_REF] Bernig | Courbures intrinsèques dans les catégories analyticogéométriques[END_REF]), Bröcker and Kuppe gave a geometric characterization of these measures using stratified Morse theory, in the more general setting of definable sets.

In [START_REF] Comte | Équisingularité réelle : nombres de Lelong et images polaires[END_REF] Comte started the study of real equisingularity by proving that the density is continuous along the strata of a Verdier stratification of a subanalytic set (see also [START_REF] Valette | Whitney conditions and Lelong number[END_REF]). The main tool to prove his result was a local Cauchy-Crofton formula for the density. He continued this work with Merle in [START_REF] Comte | Equisingularité réelle II : invariants locaux et conditions de régularité[END_REF] where a similar continuity result was established for the so-called local Lipschitz-Killing invariants (see also [START_REF] Nguyen | Whitney stratifications and the continuity of local Lipschitz-Killing curvatures[END_REF]). The tools for proving this continuity property are local linear kinematic formulas that generalize the Cauchy-Crofton formula for the density. These formulas will be explained in Section 3 but, roughly speaking, they relate the so-called polar invariants, which are mean-values of Euler characteristics of real Milnor fibres of generic projections, to the local Lipschitz-Killing invariants.

In [START_REF] Dutertre | Stratified critical points on the real Milnor fibre and integral-geometric formulas[END_REF] we also established an infinitesimal linear kinematic formula. It is slightly different from the ones of Comte and Merle, because instead of using projections, we make "infinitesimally small" translations of linear spaces. Let us recall it here because it is our main inspiration. We will use the following notations:

• s k is the volume of unit sphere S k of dimension k and b k is the volume of the unit ball B k of dimension k, • for k ∈ {0, . . . , n}, G k n is the Grassmann manifold of k-dimension linear spaces in R n equipped with the O(n)-invariant Maurer-Cartan density (see for instance [START_REF] Santaló | Integral geometry and geometric probability[END_REF], p.200), g k n is its volume, • if P is a linear subspace of R n of dimension k, S k-1 P is the unit sphere in P , • in R n , B n (x) is the closed ball of radius centered at x and S n-1 (x) is the sphere of radius centered at x, if x = 0, we simply write B n and S n-1 .

Let (X, 0) ⊂ (R n , 0) be the germ of a closed definable set. We consider the following limits:

Λ lim k (X, 0) := lim →0 Λ k (X, X ∩ B n ) b k k .
Let H ∈ G n-k n , k ∈ {1, . . . , n}, and let v be an element in S k-1 H ⊥ . For δ > 0, we denote by H v,δ the (n -k)-dimensional affine space H + δv and we set

β 0 (H, v) = lim →0 lim δ→0 Λ 0 (H δ,v ∩ X, H δ,v ∩ X ∩ B n ),
and

β 0 (H) = 1 s k-1 S k-1 H ⊥ β 0 (H, v)dv.
In [START_REF] Dutertre | Stratified critical points on the real Milnor fibre and integral-geometric formulas[END_REF] Theorem 5.5, we proved that for k ∈ {1, . . . , n}

Λ lim k (X, 0) = 1 g n-k n G n-k n β 0 (H)dH.
In view of this formula and since it is possible to make "infinitesimally small" translations of any definable set, the question that motivated us was the following: Is it possible to establish a kinematic formula for germs of closed definable sets or, in other words, can we replace the (n-k)-plane H with any germ of closed definable set? The goal of this paper is to provide a positive answer to this question.

Let us present the main results of the paper. Let (X, 0) ⊂ (R n , 0) be the germ of a closed definable set. To such a germ, we associate two sequences of real numbers: the polar invariants σ k (X, 0), k = 0, . . . , n, and the above limits Λ lim k (X, 0). Let (Y, 0) ⊂ (R n , 0) be another germ of closed definable set and let

σ(X, Y, 0) = 1 s 2 n-1 SO(n)×S n-1 lim →0 lim δ→0 + χ (X ∩ (γY + δv) ∩ B n ) dγdv.
Here SO(n) is equipped with the Haar measure dγ, normalized in such a way that the volume of SO(n) is s n-1 , S n-1 is equipped with the usual Riemanniann measure (or density) dv and SO(n) × S n-1 with the product measure dγdv. Our first infinitesimal principal kinematic formula takes the following form (Theorem 8.15):

σ(X, Y, 0) = n i=0 Λ lim i (X, 0) • σ n-i (Y, 0).
When X and Y have complementary dimensions, this gives a Bezout type formula, since the integrand of the left-hand side is a number of intersection points and the right-hand side is the product of the densities of X and Y at the origin (see Corollary 8.16). Then we set

Λ lim 0 (X, Y, 0) = 1 s 2 n-1 SO(n)×S n-1 lim →0 lim δ→0 + Λ 0 (X ∩ (γY + δv), X ∩ (γY + δv) ∩ B n ) dγdv.
In Theorem 8.17, we establish our second infinitesimal principal kinematic formula:

Λ lim 0 (X, Y, 0) = n i=0 Λ lim i (X, 0) • Λ lim n-i (Y, 0).
This formula is a corollary of Theorem 8.15 and the Gauss-Bonnet formula proved in [START_REF] Dutertre | Stratified critical points on the real Milnor fibre and integral-geometric formulas[END_REF]. Throughout the paper, we will also use the following notations and conventions:

• for v ∈ R n , the function v * : R n → R is defined by v * (y) = v, y ,

• for x ∈ R n , |x| denotes the usual Euclidean norm,

• if X ⊂ R n , X is its topological closure, X its topological interior,

• when it makes sense, vol(X) means the volume of the set X and χ c (X) its Euler characteristic for Borel-Moore homology.

The paper is organized as follows. In Section 2, we recall the notion of stratified critical points and the definition of the Lipschitz-Killing measures. We also state kinematic formulas. In Section 3, we recall the Gauss-Bonnet formula for real Milnor fibres proved by the author in [START_REF] Dutertre | Stratified critical points on the real Milnor fibre and integral-geometric formulas[END_REF], and the infinitesimal linear kinematic formulas proved by Comte [START_REF] Comte | Équisingularité réelle : nombres de Lelong et images polaires[END_REF], Comte and Merle [START_REF] Comte | Equisingularité réelle II : invariants locaux et conditions de régularité[END_REF] and the author [START_REF] Dutertre | Stratified critical points on the real Milnor fibre and integral-geometric formulas[END_REF]. Section 4 contains several topological and geometrical lemmas that will be useful in the next sections. In Section 5, we prove a new spherical kinematic formula for definable sets. Combining this formula with Hardt's theorem [START_REF] Hardt | Topological properties of subanalytic sets[END_REF][START_REF] Coste | An introduction to o-minimal geometry[END_REF][START_REF] Van Den Dries | Geometric categories and o-minimal structures[END_REF], we obtain a new kinematic formula for definable subsets of the unit ball in Section 6. We apply this formula in Section 7 to get our first principal kinematic formula for closed conic definable sets. In Section 8, we prove our first principal kinematic formula in the general case using the previous case and tangent cones, and then our second principal kinematic formula. Finally Section 9 contains two other kinematic formulas.

Stratified critical points and Lipschitz-Killing curvatures

2.1. Stratified critical points. Let X ⊂ R n be a compact definable set equipped with a finite definable Whitney stratification S = {S a } a∈A . The fact that such a stratification exists is due to Loi [START_REF] Loi | Verdier and strict Thom stratifications in o-minimal structures[END_REF] (see also [START_REF] Nguyen | A geometric proof of the existence of definable Whitney stratifications[END_REF]).

Let f : X → R be a definable function. We assume that f is the restriction to

X of a C 2 definable function F : U → R, where U is an open neighborhood of X in R n . A point p in X is a (stratified) critical point of f if p is a critical point of f |S , where S is the stratum that contains p. Definition 2.1. Let p ∈ X be an isolated critical point of f : X → R. The index of f at p is defined by ind(f, X, p) = 1 -χ (X ∩ {f = f (p i ) -δ} ∩ B n ε (p)) , where 0 < δ ε 1. If p ∈ X is not a critical point of f , we set ind(f, X, p) = 0.
Since we are in the definable setting, this index is well-defined thanks to Hardt's theorem [START_REF] Hardt | Topological properties of subanalytic sets[END_REF][START_REF] Coste | An introduction to o-minimal geometry[END_REF][START_REF] Van Den Dries | Geometric categories and o-minimal structures[END_REF].

Theorem 2.2. Assume that f : X → R has a finite number of critical points {p 1 , . . . , p s }. Then the following equality holds:

χ(X) = s i=1 ind(f, X, p i ).
Proof. See Theorem 3.1 in [START_REF] Dutertre | On the topology of semi-algebraic functions on closed semi-algebraic sets[END_REF]. When f is a Morse stratified function, this follows from [START_REF] Goresky | Stratified Morse theory[END_REF].

Lipchitz-Killing curvatures.

In this subsection, we present the Lipschitz-Killing measures of a definable set in an o-minimal structure. We describe Bröcker and Kuppe's approach [START_REF] Bröcker | Integral geometry of tame sets[END_REF].

Let X ⊂ R n be a compact definable set equipped with a finite definable Whitney stratification S = {S a } a∈A .

Let us fix a stratum S. For k ∈ {0, . . . , d S }, d S = dimS, let λ S k : S → R be defined by

λ S k (x) = 1 s n-k-1 S Tx S ⊥ ind nor (v * , X, x)σ d S -k (II x,v )dv,
where II x,v is the second fundamental form on S in the direction of v and where σ d S -k (II x,v ) is the (d S -k)-th elementary symmetric function of its eigenvalues. The index ind nor (v * , X, x) is defined as follows:

ind nor (v * , X, x) = 1 -χ X ∩ N x ∩ B n (x) ∩ {v * = v * (x) -δ} , where 0 < δ 1 and N x is a normal (definable) slice to S at x in R n . When v *
|X has a stratified Morse critical point at x, it coincides with the normal Morse index at x of a function f : R n → R such that f |X has a stratified Morse critical point at x and ∇f (x) = v. For k ∈ {d S + 1, . . . , n}, we set λ S k (x) = 0. If S has dimension n then for all x ∈ S, we put λ

S 0 (x) = • • • = λ S n-1 (x) = 0 and λ S n (x) = 1. If S has dimension 0 then ind nor (v * , X, x) = ind(v * , X, x) and we set λ S 0 (x) = 1 s n-1 S n-1 ind(v * , X, x)dv, and λ S k (x) = 0 if k > 0.
Definition 2.3. For every Borel set U ⊂ X and for every k ∈ {0, . . . , n}, we define Λ k (X, U ) by

Λ k (X, U ) = a∈A Sa∩U λ Sa k (x)dx.
These measures Λ k (X, -) are called the Lipschitz-Killing measures of X. Note that for any Borel set U of X, we have

Λ d+1 (X, U ) = • • • = Λ n (X, U ) = 0, and Λ d (X, U ) = H d (U ), where d is the dimension of X and H d is the d-th dimen- sional Hausdorff measure in R n . If X is smooth then for k ∈ {0, . . . , d}, Λ k (X, U ) is equal to 1 s n-k-1 U K d-k (x)dx.,
where

K d-k denotes the (d -k)-th Lipschitz-Killing curvature.
As in the smooth case, the measure Λ 0 (X, -) satisfies an exchange formula (see [START_REF] Bröcker | Integral geometry of tame sets[END_REF]).

Proposition 2.4. For every Borel set U ⊂ X, we have

Λ 0 (X, U ) = 1 s n-1 S n-1 x∈U ind(v * , X, x)dv.
For U = X and by Theorem 2.2, we see that a special case of this exchange formula is the Gauss-Bonnet formula Λ 0 (X, X) = χ(X).

The Lipschitz-Killing measures satisfy the kinematic formula (see [START_REF] Fu | Curvature measures of subanalytic sets[END_REF][START_REF] Bröcker | Integral geometry of tame sets[END_REF][START_REF] Bernig | Lipschitz-Killing invariants[END_REF][START_REF] Bernig | Courbures intrinsèques dans les catégories analyticogéométriques[END_REF]). We provide the group SO(n) R n of all euclideans motions of R n with the product measure dγdx, where the canonical Haar measure dγ is normalized such that vol (SO(n)) = 1.

Proposition 2.5. Let X ⊂ R n and Y ⊂ R n be two compact definable sets and let U ⊂ X and V ⊂ Y be two Borel sets. For k ∈ {0, . . . , n}, the following kinematic formula holds:

SO(n) R n Λ k (X ∩ γY, U ∩ γV )dγdx = p+q=k+n e(p, q, n)Λ p (X, U )Λ q (X, V ),
where e(p, q, n) = sp+q-nsn spsq

.

For k = 0, the above formula is called the principal kinematic formula. A particular case of the kinematic formula is the linear kinematic formula. Let A k n be the affine grassmannian of k-dimensional affine spaces in R n . It is a fibre bundle over G k n with fibre R n-k . We equip A k n with the product measure denoted by dE. Proposition 2.6. Let X ⊂ R n be a compact definabet set and let U ⊂ X be a Borel set. For k ∈ {0, . . . , n}, we have

Λ n-k (X, U ) = 1 g k n • 1 e(k, n -k, n) A k n Λ 0 (X ∩ E, X ∩ E ∩ U )dE.
In Section 5, we will consider definable subsets of the unit sphere S n-1 . For such sets, one can define spherical Lipschitz-Killing measures. These measure are defined in [START_REF] Bernig | Courbures intrinsèques dans les catégories analyticogéométriques[END_REF], Section 3 (see also [START_REF] Dutertre | Euler characteristic and Lipschitz-Killing curvatures of closed semialgebraic sets[END_REF]). Their definition is very similar to the definition of the above Lipschitz-Killing measures. For X ⊂ S n-1 and k ∈ {0, . . . , n -1}, we will denote by Λk (X, -) the k-th spherical Lispchitz-Killing measures. The spherical Lipschitz-Killing measures satisfy a Gauss-Bonnet formula ([3], Theorem 1.2) and a spherical kinematic formula ( [START_REF] Bernig | Courbures intrinsèques dans les catégories analyticogéométriques[END_REF], Theorem 4.4).

Some topological ang geometrical properties of definable sets

In this section, we review some results on the local topology and geometry of closed definable sets. Let (X, 0) be the germ of a closed definable set. For convenience, we will work with a small representative that we denote by X as well. We assume that this representative is included in a an open bounded neighborhood U of 0.

3.1. The Gauss-Bonnet formula for real Milnor fibres. We can equip X with a finite Whitney stratification S = {S α } α∈A such that 0 ∈ S α (this is possible taking a smaller representative if necessary).

Let ρ i : U → R, i = 1, 2, be two continuous definable functions of class C 2 on U \ {0}, such that ρ -1 i (0) = {0} and ρ i (x) ≥ 0 for all x ∈ X. It is wellknown that there exists i > 0 such that for 0 < ≤ i , ρ -1 i ( ) intersects X transversally in the stratified sense (see [START_REF] Dutertre | Stratified critical points on the real Milnor fibre and integral-geometric formulas[END_REF] Lemma 2.1), and that the topological type of ρ -1 i ( ) ∩ X does not depend on . Moreover, as explained by Durfee in [START_REF] Durfee | Neighborhoods of algebraic sets[END_REF], Lemma 1.8 and Corollary 3.6, there is a neighborhood Ω of 0 in R n such that for every stratum S of X, ∇(ρ 1|S ) and ∇(ρ 2|S ) do not point in opposite direction in Ω \ {0}. Applying Durfee's argument ( [START_REF] Durfee | Neighborhoods of algebraic sets[END_REF], Proposition 1.7 and Proposition 3.5), we see that ρ -1 1 ( ) ∩ X, 0 < ≤ 1 , and ρ -1 2 ( ) ∩ X, 0 < ≤ 2 , are homeomorphic. The link of X at 0, denoted by Lk(X), is the set X ∩ ρ -1 ( ), 0 < 1, where ρ : U → R is a continuous definable function of class C 2 on U \ {0}, such that ρ -1 (0) = {0} and ρ(x) ≥ 0 for all x ∈ X. We will call such a function ρ a distance function to the origin. By the above discussion, the topological type of Lk(X) does not depend on the choice of the definable distance function to the origin (actually to define the link, we do not need to assume that ρ is C 2 on U \ {0}, continuity is enough).

Let f : (X, 0) → (R, 0) be the germ of a definable function. We assume that f is the restriction to X of a C 2 definable function F : U → R. We denote by X f the set f -1 (0) and by [START_REF] Bekka | Regular stratification of subanalytic sets[END_REF][START_REF] Loi | Thom stratifications for functions definable in o-minimal structures on (R, +[END_REF], we can equip X with a definable Thom stratification V = {V β } β∈B adapted to X f . This means that {V β | V β X f } is a Whitney stratification of X \ X f and that for any pair of strata (V β , V β ) with V β X f and V β ⊂ X f , the Thom condition is satisfied.

Note that if f : (X, 0) → (R, 0) has an isolated stratified critical point at 0, where X is equipped with the above Whitney stratification S = {S α } α∈A , then the following stratification:

S α \ f -1 (0), S α ∩ (f -1 (0) \ {0}), {0} | α ∈ A , is a Thom stratification of X adapted to X f .
As explained above, there is 1 > 0 such that for 0 < ≤ 1 , ρ -1 1 ( ) intersects X f transversally. The Thom condition implies that there exists δ > 0 such that for each δ with 0 < δ ≤ δ , ρ -1 1 ( ) intersects f -1 (δ) transversally as well. Hence the set f -1 (δ) ∩ {ρ 1 ≤ } is a Whitney stratified set equipped with the following stratification:

f -1 (δ) ∩ V β ∩ {ρ 1 < }, f -1 (δ) ∩ V β ∩ {ρ 1 = } | V β X f .
Moreover, taking 1 and δ smaller if necessary, the topological types of f -1 (δ) ∩ {ρ 1 ≤ } and f -1 (δ) ∩ {ρ 1 = } do not depend on the couple ( , δ). To see this, it is enough to adapt the proof of Lemma 2.1 in [START_REF] Dutertre | On the topology of non-isolated real singularities[END_REF] to the stratified case. The same fact is true for negative values of f .

Of course, we can make the same construction with ρ 2 instead of ρ 1 . But as above, there is a neighborhood Ω of 0 in R n such that for every stratum W of X f , ∇(ρ 1|W ) and ∇(ρ 2|W ) do not point in opposite direction. Let us choose > 0 and > 0 such that {ρ 2 ≤ } {ρ 1 ≤ } ⊂ Ω . If , and δ are sufficiently small then, for every stratum V X f , ∇(ρ 1|V ∩f -1 (δ) ) and ∇(ρ 2|V ∩f -1 (δ) ) do not point in opposite direction in {ρ 1 ≤ } \ {ρ 2 < }. Otherwise, by Thom (a f )condition, we would find a point p in X f ∩ ({ρ 1 ≤ } \ {ρ 2 < }) such that either ∇(ρ 1|W )(p) or ∇(ρ 2|W )(p) vanish or ∇(ρ 1|W )(p) and ∇(ρ 2|W )(p) point in opposite direction, where W is the stratum of X f that contains p (see the proof of Lemma 3.7 in [START_REF] Dutertre | Stratified critical points on the real Milnor fibre and integral-geometric formulas[END_REF]). This is impossible if we are sufficiently close to the origin. Applying Durfee's argument mentioned above, we see that

f -1 (δ) ∩ {ρ 1 ≤ } is homeomorphic to f -1 (δ) ∩ {ρ 2 ≤ } and that f -1 (δ) ∩ {ρ 1 = } is homeomorphic to f -1 (δ) ∩ {ρ 2 = }.
The positive (resp. negative) Milnor fibre of f is the set f -1 (δ) ∩ {ρ ≤ } (resp. f -1 (-δ) ∩ {ρ ≤ }), where 0 < δ 1 and ρ is a distance function to the origin. The set f -1 (±δ) ∩ {ρ = } is the boundary of the Milnor fibre. By the previous discussion, the topological type of the positive (resp. negative) Milnor fibre and the topological type of its boundary do not depend on the choice of the definable distance function to the origin.

In [START_REF] Dutertre | Stratified critical points on the real Milnor fibre and integral-geometric formulas[END_REF], we considered a second definable function-germ g : (R n , 0) → (R, 0) and we assumed that g was the restriction to X of a C 2 definable function G : U → R. Moreover, we assumed that g satisfied the following two conditions:

• Condition (A): g : (X, 0) → (R, 0) has an isolated critical point at 0. • Condition (B): the relative polar set

Γ f,g = V β X f Γ V β f,g = V β X f x ∈ V β | rank ∇(f |V β )(x), ∇(g |V β )(x) < 2
is a 1-dimensional C 1 definable set (possibly empty) in a neighborhood of the origin.

We wrote Γ f,g = l i=1 B i , where each B i is a definable connected curve, and we considered the intersections points of Γ f,g with f -1 (δ) ∩ B n :

Γ f,g ∩ (f -1 (δ) ∩ B n ) = l i=1 B i ∩ (f -1 (δ) ∩ B n ) = p δ, 1 , . . . , p δ, r
, where 0 < |δ| 1. The points p δ, i are exactly the critical points of g on f -1 (δ) ∩ Bn . Then we set

I(δ, , g) = r i=1 ind(g, f -1 (δ), p δ, i ), I(δ, , -g) = r i=1 ind(-g, f -1 (δ), p δ, i ),
and in [START_REF] Dutertre | Stratified critical points on the real Milnor fibre and integral-geometric formulas[END_REF], Theorem 3.10, we related I(δ, , g) + I(δ, , -g), with 0 < |δ| 1, to the topology of the Milnor fibre and its boundary.

Let us give now a new characterization of I(δ, , g) and I(δ, , -g) independent on δ and . Let us fix a connected component B of Γ f,g . We can assume that f is stricly increasing on B and we put B ∩ f -1 (δ) = {p δ } for δ > 0.

Lemma 3.1. There exists δ 0 > 0 such that for 0 < δ ≤ δ 0 , the function δ → ind(g, f -1 (δ), p δ ) is constant on ]0, δ 0 ].

A = {x ∈ X | ∃p ∈ B such that f (x) = f (p) and g(x) ≤ g(p)} .
It is a definable subset of X. Let ρ : A ∩ (O \ B) → R 2 be the mapping defined by ρ(x) = (f (x), d(x)). By Hardt's theorem [START_REF] Hardt | Topological properties of subanalytic sets[END_REF][START_REF] Coste | An introduction to o-minimal geometry[END_REF][START_REF] Van Den Dries | Geometric categories and o-minimal structures[END_REF], there is a partition of ]0, +∞[×]0, +∞[ into finitely many definable sets such that ρ is trivial over each of this set. Let us denote by ∆ the union of the members of this partition which have dimension less than or equal to 1. By Hardt's theorem again, the set

{ν ∈]0, +∞[ | ∆ ∩ ({ν}×]0, +∞[) has dimension 1}
is finite. For ν > 0, the function r(ν) = inf{ | (ν, ) ∈ ∆} is definable and by the previous remark, there is ν 1 > 0 such that r(ν) > 0 for 0 < ν < ν 1 . Hence by the Monotonicity Theorem (see [START_REF] Coste | An introduction to o-minimal geometry[END_REF], Theorem 2.1 or [START_REF] Van Den Dries | Geometric categories and o-minimal structures[END_REF], 4.1), there is 0 < δ 0 < ν 1 such that r is continuous, monotone and strictly positive on ]0,

δ 0 ]. Moreover the function (δ, ) → χ (A ∩ {f = δ} ∩ {d = }) is constant on {(δ, ) | 0 < δ < δ 0 , 0 < < r(δ)}.
But, by Lemma 3.1 in [START_REF] Dutertre | On the topology of non-isolated real singularities[END_REF] and the above discussion on the topology of the link, we have

ind(g, f -1 (δ), p δ ) = 1 -χ {g ≤ g(p δ )} ∩ {f = δ} ∩ {d = } = 1 -χ (A ∩ {f = δ} ∩ {d = }) . We conclude that the function δ → ind(g, f -1 (δ), p δ ) is constant on ]0, δ 0 ].
Of course, a similar result holds for negative values of f . By the general Lojasiewicz inequality (see [START_REF] Bröcker | Integral geometry of tame sets[END_REF], Corollary 1.5.2), there exists a continuous definable function ψ

: (R, 0) → (R, 0) such that |p| ≤ ψ(f (p)) for p ∈ B. Moreover ψ is of class C 2 in an open neighborhood of 0 and ψ(u) > 0 for u > 0. Let us fix > 0 small. If 0 < δ < ψ -1 ( 4 ) then |p| ≤ 4 for p ∈ B ∩ f -1 (δ).
Since Γ f,g consists of a finite number of branches, we can conclude that for > 0 there exists δ 0 > 0 such that for 0

< |δ| ≤ δ 0 , Γ f,g ∩ f -1 (δ) ⊂ B n 4 , and so Γ f,g ∩ (f -1 (δ) ∩ B n ) = Γ f,g ∩ f -1 (δ). With the above notation, this means that p δ, i = p δ i for 0 < |δ| 1 and i ∈ {1, . . . , r}. For i ∈ {1, . . . , r}, let τ i (g) (resp. τ i (-g)) be the value that the function δ → ind(g, f -1 (δ), p δ i ) (resp. ind(-g, f -1 (δ), p δ i )
) takes close to the origin. We deduce the following relations: lim

→0 lim δ→0 + I(δ, , g) + I(δ, , -g) = i | f >0 on Bi τ i (g) + τ i (-g), lim →0 lim δ→0 - I(δ, , g) + I(δ, , -g) = i | f <0 on Bi τ i (g) + τ i (-g).
Of course, the same study can be done with another definable distance function to the origin and so, the two limits lim →0 lim δ→0 + I(δ, , g) + I(δ, , -g) and lim →0 lim δ→0 -I(δ, , g) + I(δ, , -g) do not depend on the distance function to the origin chosen to define the Milnor fibre of f . Applying this study to a linear form v * , where v is generic in S n-1 , we established in [START_REF] Dutertre | Stratified critical points on the real Milnor fibre and integral-geometric formulas[END_REF], Theorem 4.5, an infinitesimal Gauss-Bonnet formula for the Milnor fibre of f . We will use only this formula for functions with an isolated stratified critical point at 0. Namely if X is equipped with a Whitney stratification for which f : (X, 0) → (R, 0) has an isolated stratified critical point at 0, then ( [START_REF] Dutertre | Stratified critical points on the real Milnor fibre and integral-geometric formulas[END_REF], Corollary 4.6)

lim →0 lim δ→±0 Λ 0 f -1 (δ), f -1 (δ) ∩ B n = lim →0 lim δ→±0 χ f -1 (δ) ∩ B n - 1 2 χ Lk(X f ) - 1 2s n-1 S n-1 χ Lk(X f ∩ {v * = 0}) dv.
The proof of this Gauss-Bonnet formula relies on the following exchange formula:

lim →0 lim δ→±0 Λ 0 f -1 (δ), f -1 (δ) ∩ B n = 1 2s n-1 S n-1 lim →0 lim δ→±0 I(δ, , v * ) + I(δ, , -v * ) dv.
But we have explained above that lim →0 lim δ→0 I(δ, , v * ) + I(δ, , -v * ) does not depend on the choice of the distance function to the origin used to define the Milnor fibre of f . Therefore the relations proved in [START_REF] Dutertre | Stratified critical points on the real Milnor fibre and integral-geometric formulas[END_REF], Theorem 4.5 and Corollary 4.6, are also valid if we replace the usual euclidian distance function by any definable distance function to the origin. This remark will be important in the next sections.

3.2. Linear kinematic formulas for germs of closed definable sets. Let us recall the definition of the polar invariants [START_REF] Comte | Equisingularité réelle II : invariants locaux et conditions de régularité[END_REF]. Let k ∈ {1, . . . , n} and let P ∈ G k n . We denote by π P : X → P the orthogonal projection on P . For P generic in G k n , Comte and Merle established the existence of an open and dense definable germ (K P , 0) ⊂ (P, 0) such that, if K P = ∪ N P i=1 K P i denotes its decomposition into connected components, then the function

K P i → χ P i := lim →0 lim y∈K P i y→0 χ π -1 P (y) ∩ X ∩ B n
is well-defined. Then they set the following definition:

Definition 3.2. Let k ∈ {1, . . . , n}. The polar invariant σ k (X, 0) is defined by σ k (X, 0) = 1 g k n G k n N P i=1 χ P i • Θ(K p i , 0)dP.
We set σ 0 (X, 0) = 1.

In [START_REF] Comte | Equisingularité réelle II : invariants locaux et conditions de régularité[END_REF] the authors defined another sequence of invariants attached to X, called the local Lipschitz-Killing invariants.

Definition 3.3. Let k ∈ {0, . . . , n}. The local Lipschitz-Killing invariant Λ loc k (X, 0) is defined by Λ loc k (X, 0) = lim →0 Λ k (X ∩ B n , X ∩ B n ) b k k .
We note that Λ loc 0 (X, 0) = 1. Then Comte and Merle proved linear kinematic formulas that relate the local Lipschitz-Killing invariants to the polar invariants. Theorem 3.4 ([9], Theorem 3.1). For any germ (X, 0) ⊂ (R n , 0) of definable closed set, we have

   Λ loc 1 (X, 0) . . . Λ loc n (X, 0)    =      1 m 2 1 . . . m n 1 0 1 . . . m n 2 . . . . . . . . . . . . 0 0 . . . 1      •    σ 1 (X, 0) . . . σ n (X, 0)    ,
where

m j i = bj bj-ibi j i - bj-1 bj-1-ibi j-1 i , for i + 1 ≤ j ≤ n. If dimX = d then σ d+1 (X, 0) = • • • = σ n (X, 0) = 0 and one recovers the local Cauchy-Crofton formula Λ loc d (X, 0) = σ d (X, 0)
, previously proved by Comte [START_REF] Comte | Équisingularité réelle : nombres de Lelong et images polaires[END_REF]. In [START_REF] Dutertre | Euler characteristic and Lipschitz-Killing curvatures of closed semialgebraic sets[END_REF], we also studied the asymptotic behavior of the Lipschitz-Killing measures. For k = 0, . . . , n, we considered the limits

Λ lim k (X, 0) := lim →0 Λ k (X, X ∩ B n ) b k k ,
and we showed the following theorem:

Theorem 3.5 ([14], Theorem 5.1). For any germ (X, 0) ⊂ (R n , 0) of definable closed set, we have

Λ lim 0 (X, 0) = 1 - 1 2 χ(Lk(X)) - 1 2g n-1 n G n-1 n χ(Lk(X ∩ H))dH.
Furthermore for k ∈ {1, . . . , n -2}, we have

Λ lim k (X, 0) = - 1 2g n-k-1 n G n-k-1 n χ(Lk(X ∩ H))dH + 1 2g n-k+1 n G n-k+1 n χ(Lk(X ∩ L))dL,
and:

Λ lim n-1 (X, 0) = 1 2g 2 n G 2 n χ(Lk(X ∩ H))dH, Λ lim n (X, 0) = 1 2g 1 n G 1 n χ(Lk(X ∩ H))dH.
As a corollary, we obtained: Corollary 3.6 ([14], Corollary 5.2). For any germ (X, 0) ⊂ (R n , 0) of definable closed set, the equality 1 = n k=0 Λ lim k (X, 0) holds. We note that Λ lim k (X, 0) differs from Λ loc k (X, 0), because the link of X does not have any contribution in the computation of Λ lim k (X, 0). In [START_REF] Dutertre | Stratified critical points on the real Milnor fibre and integral-geometric formulas[END_REF], we continued our study of the Lipschitz-Killing measures and established a local linear kinematic formula for Λ lim k (X, 0), which was our inspiration for the present work. Let H ∈ G n-k n , k ∈ {1, . . . , n}, and let v be an element in S k-1 H ⊥ . For δ > 0, we denote by H v,δ the (n -k)-dimensional affine space H + δv and we set

β 0 (H, v) = lim →0 lim δ→0 Λ 0 (H δ,v ∩ X, H δ,v ∩ X ∩ B n ). Then we set β 0 (H) = 1 s k-1 S k-1 H ⊥ β 0 (H, v)dv.
Theorem 3.7 ([15], Theorem 5.5). For k ∈ {1, . . . , n}, we have

Λ lim k (X, 0) = 1 g n-k n G n-k n β 0 (H)dH.
We also proved local linear kinematic formulas that relate the limits Λ lim k (X, 0) to the polar invariants. Theorem 3.8 ([15], Theorem 5.6). For k ∈ {0, . . . , n -1}, we have

Λ lim k (X, 0) = σ k (X, 0) -σ k+1 (X, 0). Furthermore, we have Λ lim n (X, 0) = σ n (X, 0).

Some preliminary topological and geometrical results

Let (X, 0) and (Y, 0) be two germs of closed definable sets in R n . For convenience, we will work with two representatives of these germs that we denote by X and Y as well. We assume that these representatives X and Y are included in an open neighborhood U of 0.

4.1.

A Gauss-Bonnet formula. Let {S i } l i=0 be a Whitney stratification of X, where S 0 = {0} and 0 ∈ S i for i ∈ {1, . . . , l}. Similarly let {T j } m j=0 be a Whitney stratification of Y , where T 0 = {0} and 0 ∈ T j for j = {1, . . . , m}. We assume each stratum to be connected. We introduce the following condition:

• Condition (1): for i ∈ {1, . . . , l} and for j ∈ {1, . . . , m}, S i and T j intersect transversally (the case

S i ∩ T j = ∅ is possible). If X and Y satisfy Condition (1) then X ∩ Y admits a Whitney stratification X ∩ Y = r k=0 R k , where R 0 = {0} and each R k is a connected component of an intersection S i ∩ T j , (i, j) ∈ {1, . . . , l} × {1, . . . , m}.
Let X ⊂ R n+1 be the following definable set:

X = (x, t) ∈ R n+1 | x ∈ X .
It is included and closed in U × R. Let v ∈ S n-1 and let Y v ⊂ R n+1 be the following definable set:

Y v = (y, t) ∈ R n+1 | ∃y ∈ Y such that y = y + tv .

It is included and closed in the open set

U v = (u, t) ∈ R n+1 | ∃u ∈ U such that u = u + tv .
It is well-known (see [START_REF] Goresky | Stratified Morse theory[END_REF]) that we can equip X with a Whitney stratification in the following way:

X = l i=0 S i where S i = {(x, t) ∈ R n+1 | x ∈ S i }. Similarly we can consider the following partition of Y v : Y v = m j=0 (T j ) v where (T j ) v = (y, t) ∈ R n+1 | ∃y ∈ T j such that y = y + tv . Lemma 4.1. The partition Y v = m j=0 (T j ) v gives a Whitney stratification of Y v .
Proof. With obvious notations, the partition Y = m j=0 T j induces a Whitney stratification of Y . Let φ : U × R → U v be defined by φ(u, t) = (u + tv, t). Then φ is a diffeomorphism, φ( Y ) = Y v and φ( T j ) = (T j ) v for j ∈ {0, . . . , m}. This gives the result for Whitney's conditions are invariant by C 1 -diffeomorphisms.

From now on, we will focus on the definable set X ∩ Y v . Let us denote if by Z v . It is included and closed in the open set (U × R) ∩ U v . We introduce the following second condition:

• Condition (2): for i ∈ {0, . . . , l} and j ∈ {0, . . . , m}, the strata S i and (T j ) v intersect transversally outside (0, 0). If v satisfies Condition (2) then Z v admits a Whitney stratification Z v = q l=0 Q l where each Q l is a connected component of an intersection S i ∩ (T j ) v . We note that necessarly S 0 ∩ (T 0 ) v = {(0, 0)} and that we can put Q 0 = {(0, 0)}. Lemma 4.2. Assume that X and Y satisfy Condition [START_REF] Bekka | Regular stratification of subanalytic sets[END_REF] and that v satisfies Condition [START_REF] Bernig | Lipschitz-Killing invariants[END_REF]. Then the function

t |Zv : Z v → R (y, t) → t
has an isolated stratified critical point at (0, 0).

Proof. Let Q be a stratum of Z v different from {(0, 0)}. Since the critical points of t |Q lie in {t = 0}, we can suppose that Q is a connected component of S i ∩ (T j ) v with i = 0 and j = 0. Let us prove that {t = 0} intersects S i ∩ (T j ) v transversally. If it is not the case, then there is a point p in S i ∩ (T j ) v ∩ {t = 0} such that T p ( S i ∩ (T j ) v ) ⊂ R n . But it is not difficult to check that {t = 0} intersects S i and (T j ) v transversally, so T p S i = T p S i ∩ R n and T p T j = T p (T j ) v ∩ R n . Moreover, S i and T j intersect transversally and so T p (S i ∩ T j ) = T p S i ∩ T p T j . Similarly, T p ( S i ∩ (T j ) v ) = T p S i ∩ T p (T j ) v . We get that T p (S i ∩ T j ) = T p ( S i ∩ (T j ) v )
. This is not possible, for dim S i = dimS i + 1, dim (T j ) v = dimT j + 1 and these two intersections are transverse in R n and R n+1 .

We can apply Corollary 4.6 in [START_REF] Dutertre | Stratified critical points on the real Milnor fibre and integral-geometric formulas[END_REF] to t and Z v .

Corollary 4.3. Assume that X and Y satisfy Condition [START_REF] Bekka | Regular stratification of subanalytic sets[END_REF] and that v satisfies Condition (2). Then we have

lim →0 lim δ→±0 Λ 0 Z v ∩ {t = δ}, Z v ∩ {t = δ} ∩ B n+1 = lim →0 lim δ→±0 χ Z v ∩ {t = δ} ∩ B n+1 - 1 2 χ (Lk(X ∩ Y )) - 1 2s n-1 S n-1 χ (Lk(X ∩ Y ∩ {u * = 0}) du.
Proof. We just have to show that

1 2s n-1 S n-1 χ (Lk(X ∩ Y ∩ {u * = 0}) du = 1 2s n S n χ (Lk(X ∩ Y ∩ {u * = 0}) du.
But since X ∩ Y is included in R n , the method given in the proof of Corollary 5.1 [START_REF] Dutertre | A Gauss-Bonnet formula for closed semi-algebraic sets[END_REF] applies here.

Let us go back now to the sets X and Y . We denote the definable set X ∩(Y +δv) by Z v,δ . Lemma 4.4. There exists 0 > 0 such that for 0 < ≤ 0 , there exists δ > 0 such that for 0 < δ ≤ δ , the topological type of Z v,δ ∩ B n does not depend on the choice of the couple ( , δ). Moreover for 0 < δ

1, Z v,δ ∩ B n and Z v ∩ {t = δ} ∩ B n+1 are homeomorphic. Proof. Let ρ(x, t) = x 2 1 + • • • + x 2 n + t. Then Z v,δ ∩ B n is homeomorphic to Z v ∩ {t = δ} ∩ {ρ ≤ + δ}. Let π : Z v ∩ {t ≥ 0} → R
be the mapping defined by π(x, t) = (ρ(x, t), t) and let ∆ ⊂ R 2 be its (stratified) discriminant. It is a definable curve in a neighborhood of 0 ∈ ∆ in R + × R + . The following function

r : R + → R + ν → inf{t | (ν, t) ∈ ∆}
is a definable function defined in a neighborhood of 0. Note that r(0) = 0 and r(ν) > 0 for ν > 0 close to 0. By the Monotonicity Theorem (see Theorem 2.1 in [START_REF] Coste | An introduction to o-minimal geometry[END_REF] or [START_REF] Van Den Dries | Geometric categories and o-minimal structures[END_REF], 4.1), we can assume that r is continuous and increasing on a small interval ]0, ν 0 [. Let ( , δ) be a couple such that 0 < < ν 0 , 0 < δ < r( ) and Z v ∩ {t = δ} ∩ {ρ ≤ } has the topological type of the Milnor fibre of t |Zv . By taking δ smaller if necessary, we can assume that + δ < ν 0 . Since r is increasing, δ < r( ) < r( + δ) and so

Z v ∩ {t = δ} ∩ {ρ ≤ + δ} is homeomorphic to Z v ∩ {t = δ} ∩ {ρ ≤ }.
We conclude with the results of Section 3.

A similar result is true for negative values of t replacing

x 2 1 + • • • + x 2 n + t with x 2 1 + • • • + x 2 n -t.
We can state the infinitesimal Gauss-Bonnet formula for Z v,δ . Lemma 4.5. Assume that X and Y satisfy Condition [START_REF] Bekka | Regular stratification of subanalytic sets[END_REF] and that v satisfies Condition (2). Then we have

lim →0 lim δ→±0 Λ 0 (Z v,δ , Z v,δ ∩ B n ) = lim →0 lim δ→±0 χ (Z v,δ ∩ B n ) - 1 2 χ (Lk(X ∩ Y )) - 1 2s n-1 S n-1 χ (Lk(X ∩ Y ∩ {u * = 0})) du.
Proof. Let i : R n → R n+1 , x → (x, δ). Since i is a definable isometry, by Theorem 5.0 in [START_REF] Fu | Curvature measures of subanalytic sets[END_REF] or Proposition 9.2 in [START_REF] Bröcker | Integral geometry of tame sets[END_REF], we have

Λ n 0 (Z v,δ , Z v,δ ∩ B n ) = Λ n+1 0 (Z v ∩ {t = δ}, Z v ∩ {t = δ} ∩ {ρ ≤ + δ}) .
Here we suppose that δ > 0, Λ n 0 (resp. Λ n+1 0

) stands for the Gauss-Bonnet measure in R n (resp. R n+1 ) and ρ(x, t) = x 2 1 + • • • + x 2 n + t As explained in Section 3, for u generic in S n and for > 0 sufficiently small, there exists δ ,u such that for 0 < δ ≤ δ ,u , the critical points of u * and -u * in Z v,δ ∩ {t = δ} actually lie in Z v,δ ∩ {t = δ} ∩ {ρ ≤ 4 }, hence there are not in Z v,δ ∩ {t = δ} ∩ { ≤ ρ ≤ + δ}. Thanks to this observation, we can conclude that lim

→0 lim δ→0 Λ n+1 0 (Z v ∩ {t = δ}, Z v ∩ {t = δ} ∩ {ρ ≤ + δ}) = lim →0 lim δ→0 Λ n+1 0 (Z v ∩ {t = δ}, Z v ∩ {t = δ} ∩ {ρ ≤ }) .
It is enough to apply Corollary 4.3 and the comments of Section 3 on the choice of the distance function to the origin to get the result.

4.2.

A useful lemma. We continue this section with a remark. Instead of translating Y , we can translate X, intersect this translated set with Y and obtain another Milnor fibre Y ∩ (X + δv) ∩ B n , 0 < |δ| 1.

Lemma 4.6. There exists 0 > 0 such that for 0 < ≤ 0 , there exists δ > 0 such that 0

< δ ≤ δ , X ∩ (Y + δv) ∩ B n and Y ∩ (X -δv) ∩ B n are homeomorphic. Proof. Let ρ v (x, t) = (x 1 -tv 1 ) 2 + • • • + (x n -tv n ) 2 +t.
By the results of Section 3, we know that there exists 0 > 0 such that for 0 < ≤ 0 , there exists δ > 0 such that for 0 < δ ≤ δ , the topological type of Z v ∩ {t = δ} ∩ {ρ v ≤ } does not depend on the couple ( , δ) and is the topological type of the positive Milnor fibre of t |Zv . On the other hand, the set X ∩ (Y + δv) ∩ (B n + δv) is homeomorphic to

Z v ∩ {t = δ} ∩ {ρ v ≤ + δ}.
The same method as the one used in Lemma 4.4 shows that for 0 < ≤ 0 and 0 < δ ≤ δ small enough,

Z v ∩ {t = δ} ∩ {ρ v ≤ + δ} is homeomorphic to Z v ∩ {t = δ} ∩ {ρ v ≤ }. We conclude that Z v,δ ∩ B n is homeomorphic to Z v,δ ∩ (B n + δv) for 0 < δ 1. But X ∩ (Y + δv) ∩ (B n + δv) -δv = (X -δv) ∩ Y ∩ B n .
4.3. Genericity of Conditions ( 1) and ( 2). We prove the genericity of Conditions ( 1) and (2). To prove the genericity of Condition (1), we need some auxiliary lemmas.

Lemma 4.7. Let x ∈ R n be a non-zero vector. We have

Hx | H ∈ M n (R) such that t H = -H = x ⊥ .
Proof. It is clear that if H is an antisymmetric matrix, then Hx ∈ x ⊥ . Let us write x = (x 1 , . . . , x n ) and let a = (a 1 , . . . , a n ) ∈ x ⊥ . Since x = (0, . . . , 0) then there exists k such that x k = 0. Then we can construct H = (h ij ) in the following way:

h kj = - a j x k , h jk = -h kj for j = k,
and putting h ij = 0 for the other coefficients. Then H is antisymmetric and Hx = a.

Lemma 4.8. Let f : R n → R k with 1 ≤ k ≤ n -1 be a C 1 mapping and let F be the mapping defined by

F : M n (R) → S n (R) × R k A → ( t AA, f (Ax)),
where x is a non-zero vector.

If A ∈ SO(n) and Df (Ax) |(Ax) ⊥ : (Ax) ⊥ → R k is surjective then DF (A) is a surjection. Proof. We have DF (A)(H) = ( t AH + t HA, Df (Ax)(Hx)). Let (Y, α) ∈ S n (R)×R k , we have that t A 1 2 AY + t 1 2 AY A = Y. Let β = Df (Ax)( 1 2 AY x).
We have to find H such that t AH + t HA = 0 and Df (Ax)(Hx) = α -β. Since Df (Ax) |(Ax) ⊥ is a surjection, by the previous lemma, there exists an antisymmetric matrix L such that Df (Ax)(LAx) = α -β. We take H = LA. Lemma 4.9. Let T ⊂ R n be a C 1 definable submanifold of dimension d such that 0 ∈ T . Then there exists a neighborhood U T of 0 such that for

x ∈ T ∩ (U T \ {0}), dim(T x T ∩ x ⊥ ) ≤ d -1.
Proof. If it is not the case, then there exists an injective C 1 definable map µ : [0, ν) → T such that µ(0) = 0 and for t = 0, T µ(t) T ⊂ µ(t) ⊥ , hence µ(t) ⊥ T µ(t) T . Since µ (t) ∈ T µ(t) T , we get that µ(t), µ (t) = 0. This implies that µ(t), µ(t) = 0 and that |µ(t)| = 0, which is not possible. Lemma 4.10. There exists a definable subset Σ X,Y ⊂ SO(n) of positive dimension such that for γ / ∈ Σ X,Y , X and γY satisfy Condition (1).

Proof. We note first that {γT j } m j=0 is a Whitney stratification of γY . Let S = {0} be a stratum of X and let T = {0} be a stratum of Y . We have to prove that there exists a definable subset Σ S,T ⊂ SO(n) of positive codimension such that S and γT intersect transversally in a neighborhood of 0. If dimS = n or dimT = n, there is nothing to prove so we can assume that e := dimS ≤ n -1 and d := dimT ≤ n -1. Let M be the following definable set:

M = (p, γ) ∈ (U \ {0}) × O(n) | p ∈ S ∩ γ -1 T , where U is an open definable neighborhood of 0, included in U ∩ U T and such that γU ⊂ U for all γ ∈ O(n). Let us prove that M is a definable submanifold of R n × M n (R). Let (p, γ) be a point in M . There is an open neighborhood V of (p, γ) in R n × M n (R) such that in V, M is defined G(x, A) = (0, I n , 0), where G : R n × M n (R) → R n-e × S n (R) × R n-d (x, A) → (g(x), t AA, f (Ax)),
and where g is a definable submersion such that S is locally defined by g(x) = 0 in a neighborhood of p and f is a definable submersion such that T is locally defined by f (x) = 0 in a neighborhood of γp.

Since p belongs to U , γp ∈ U T ∩ T and so dim(T γp T ∩ (γp) ⊥ ) ≤ d -1, which implies that Df (γp) |(γp) ⊥ : (γp) ⊥ → R n-d is a surjection. By Lemma 4.8, the mapping

F : M n (R) → S n (R) × R n-d A → ( t AA, f (Ap))
is a submersion at γ. Therefore the submatrix of the jacobian matrix of G at (p, γ) formed by the partial derivatives of F with respect A has maximal rank. But the submatrix of the jacobian matrix of G at (p, γ) formed by the partial derivatives of g with respect to x has also maximal rank. We conclude that G is a submersion at (p, γ) and that M is a definable submanifold of dimension

n + n 2 -2n - n(n + 1) 2 + (e + d) = n(n -1) 2 + (e + d) -n.
Let π : M → O(n) be the natural projection. By Sard's theorem (see [START_REF] Bochnak | Géométrie algébrique réelle[END_REF]), its discriminant ∆ is a definable subset of positive codimension. Let T : O(n) → O(n) be the definable diffeomorphism given by T (A) = t A. It is enough to take Σ S,T = T (∆) ∩ SO(n).

Lemma 4.11. Assume that X and Y satisfy Condition [START_REF] Bekka | Regular stratification of subanalytic sets[END_REF]. There exists a definable subset Γ X,Y ⊂ S n-1 of positive codimension such that for v / ∈ Γ X,Y , v satisfies Condition (2).

Proof. Let S be a stratum of X and let T be a stratum of Y . We have to prove that there exists a definable subset Γ S,T ⊂ S n-1 of positive codimension such that S and T v intersect transversally outside (0, 0), in a neighborhood of (0, 0).

If S = {0} and T = {0}, then S ∩ T v = {(0, 0)} and there is nothing to prove. If dimS = n or dimT = n, there is nothing to prove neither. Let us treat first the case 0 < e := dimS ≤ n -1 and 0 < d := dimT ≤ n -1. Let M be the following definable set:

M = (p, τ, ν) ∈ U × R × (R n \ {0}) | (p, τ ) ∈ S ∩ T ν .
Let us prove that M is a definable submanifold. Let (p, τ, ν) be a point in M . There is an open neighborhood V of (p, τ, ν) ∈ U × R × (R n \ {0}) such that in V, M is defined by g(x) = 0 and f (x-tv) = 0, where g is a definable submersion such that S is locally defined by g(x) = 0 in a neighborhood of p and f is a definable submersion such that T is locally defined by f (x) = 0 in a neighborhood of p -τ ν. It is easy to check that the Jacobian matrix of the mapping (g, f ) has maximal rank at (p, τ, ν) if τ = 0. If τ = 0 this is also the case by Condition (1). Therefore M is a definable submanifold of dimension 2n + 1 -2n + (e + d) = (e + d) + 1. Let π : M → R n be the projection π(p, τ, ν) = ν. By Sard's theorem (see [START_REF] Bochnak | Géométrie algébrique réelle[END_REF]), its discriminant ∆ is a definable subset of positive codimension. We take Γ S,T = S n-1 \ ∆. The remaining two cases are proved with the same method. 4.4. The link of Z v ∩ {t ≥ 0}. We study the link of the set Z v ∩ {t ≥ 0}. We still assume that X and Y satisfy Condition [START_REF] Bekka | Regular stratification of subanalytic sets[END_REF] and that v satisfies Condition (2). For (x, t) ∈ R n+1 , we set ω(x, t) = |x|. Let 0 > 0 be such that B n 0 ⊂ U . We set

Γ Y = ( 1 u y, u) | y ∈ Y, u ∈]0, 0 [ ⊂ R n+1 .
We recall that the tangent cone of Y at 0 is

C 0 Y = Γ Y ∩ (R n × {0}).
It is a closed and conic definable set in R n .

Lemma 4.12. If v / ∈ (-C 0 (Y )) ∩ S n-1 then there exist v > 0 and a > 0 such that the inclusion

Y v ∩ {t ≥ 0} ∩ B n+1 v ⊂ (x, t) ∈ R n+1 | |x| ≥ at ∩ B n+1 v holds.
Proof. If it is not the case then we can find a sequence of points (x n , t n ) n∈N in Y v \ {t = 0} such that (x n , t n ) → (0, 0) and lim n→+∞ |xn| tn = 0. We have

|x n -t n v| 2 = |x n | 2 + t 2 n -2t n x n , v
x n -t n v |x n -t n v| = -v, which implies that -v belongs to C 0 (Y ) ∩ S n-1 . Corollary 4.13. If v / ∈ (-C 0 (Y ))∩S n-1 then for 0 < 1, the link of Z v ∩{t ≥ 0} is homemorphic to Z v ∩ {t ≥ 0} ∩ {(x, t) | |x| = }. Proof. It is enough to show the implication (y 0 , t 0 ) ∈ Y v ∩ {t ≥ 0} ∩ B n+1 v ω(y 0 , t 0 ) = 0 ⇒ (y 0 , t 0 ) = (0, 0).
But if ω(y 0 , t 0 ) = 0 then |y 0 | = 0 and, by the previous lemma, |y 0 | ≥ at 0 for some a > 0.

If dimC 0 (Y ) ≤ n -1 then the previous corollary holds for almost all v in S n-1 since dimC 0 (Y ) ∩ S n-1 < n -1.

In the rest of this subsection, we assume that dimC 0 (Y ) = n. This implies that dimY = n. Let us denote by Y the union of the strata of Y of dimension less than or equal to n -1. It is a closed definable set of dimension less than or equal to n -1, if not empty. We need auxiliary lemmas. Lemma 4.14. Let S ⊂ R n be a definable open subset such that 0 ∈ S and that

dimC 0 S = n. If v is in C 0 S \ C 0 (S \ S) then there is α > 0 such that ]0, α] • v ⊂ S.
Proof. We note that C 0 S \ C 0 (S \ S) is not empty because dimC 0 (S) = n. Let v ∈ C 0 S \ C 0 (S \ S) (note that necessarly v = 0). Let us suppose that for all α > 0, ]0, α] • v is not included in S. Hence we can construct a sequence (z n ) n∈N in the complement c S of S such that (z n ) tends to 0 and zn |zn| = v. This implies that v belongs to C 0 ( c S). Since v belongs to C 0 (S) as well, there exists a sequence of points (x n ) n∈N in S such that xn |xn| tends to v. Let a ∈ S and let b ∈ c S. Let [a, b] be the segment with extremities a and b, i.e. For each n ∈ N, let y n be a point in [x n , z n ] ∩ (S \ S). If there is a subsequence y τ (n) such that y τ (n) = 0, then

x τ (n) |x τ (n) | = -v, which is not possible for x τ (n) |x τ (n) | tends to v.
Therefore we can assume that y n = 0 for n ∈ N, and write

y n |y n | = λ n |z n | |y n | • v + (1 -λ n )|x n | |y n | • x n |x n | ,
where λ n ∈ [0, 1]. For simplicity we rewrite this equality in the following way:

y n |y n | = α n • v + β n • x n |x n | , with α n , β n ≥ 0. Since v, xn |xn| → 1, there is n 0 such that for n ≥ n 0 , v, xn |xn| ≥ 1 2 . This implies that for n ≥ n 0 , 0 ≤ α 2 n + β 2 n + α n β n ≤ 1
and so the sequences (α n ) n∈N and (β n ) n∈N are bounded. Therefore, taking a subsequence if necessary, we can assume that α n tends to α ≥ 0 and β n tends to β ≥ 0. Hence yn |yn| tends to (α + β)v, where α + β = 1 for the limit of yn |yn| is a unit vector. We see that v belongs to C 0 (S \ S), which is not possible by hypothesis. We conclude that there is α > 0 such that ]0, α] • v ⊂ S. Lemma 4.15. Let W ⊂ R n be a closed definable set equipped with a Whitney stratification. Suppose that 0 ∈ W and that 0 lies in a stratum of dimension greater than or equal to 1. Let g : W → R be a definable function, restriction of a C 2 definable function, such that 0 is not a stratified critical point of g. Let f : W ∩{g ≤ 0} → R be a definable function, restriction of a C 2 definable function, such that f (0) = 0 and 0 is a local strict maximum of f . Then ind(f, W ∩ {g ≤ 0}, 0) = 0.

Proof. By Lemma 3.3 in [START_REF] Dutertre | On the topology of non-isolated real singularities[END_REF], χ (Lk(W ∩ {g ≤ 0})) = 1. Since -f |W ∩{g≤0} is a distance function to the origin, we can write

ind(f, W ∩ {g ≤ 0}, 0) = 1 -χ (W ∩ {g ≤ 0} ∩ {-f ≤ } ∩ {f = -δ}) , with 0 < δ < 1. But W ∩ {g ≤ 0} ∩ {-f ≤ } ∩ {f = -δ} is homeomorphic to the link of W ∩ {g ≤ 0} at 0. Let us choose v in (-C 0 (Y )) ∩ S n-1 such that v / ∈ (-C 0 (Y )) ∩ S n-1
. By Lemma 4.12, there exists v > 0 and a > 0 such that

Y v ∩ {t ≥ 0} ∩ B n+1 v ⊂ {(x, t) | |x| ≥ at} ∩ B n+1 v .
Lemma 4.16. Under these assumptions, we have

χ X ∩ (Y + 2 a v) ∩ B n = 1, for 0 < 1. Proof. For 0 < 1, we set r = 2 + 2 a 2 . The set X ∩ (Y + 2 a ) ∩ B n is the intersection Z v ∩ {t = 2 a } ∩ B n+1 r . If (x 0 , t 0 ) lies in Z v ∩ {t ≥ 2 a } ∩ B n+1 r then |x 0 | 2 ≤ 2 + 2 a 2 -t 2 0 and |x0| 2 t 2 0 ≤ 2 t 2 0 ≤ a 2 2
. Therefore if is sufficiently small,

Z v ∩ {t ≥ 2 a } ∩ B n+1 r ∩ Y v = ∅.
The function t |Zv has an isolated stratified critical value at 0. If is small enough, the stratified critical points of t |Zv∩{t> 2

a }∩B n+1 r lie in Z v ∩ {t > 2 a } ∩ S n r . But Y v does not intersect Z v ∩ {t > 2 a } ∩ B n+1 r , so Z v ∩ {t > 2 a } ∩ B n+1 r (resp. Z v ∩ {t > 2 a } ∩ S n r ) is stratified by strata of the form S ∩ T v ∩ {t > 2 a } ∩ B n+1 r (resp. S ∩ T v ∩ {t > 2 a } ∩ S n r ), where S is a stratum of X and T is a stratum of Y of dimension n. This means that Z v ∩ {t > 2 a } ∩ B n+1 r (resp. Z v ∩ {t > 2 a } ∩ S n r ) is stratified by open subsets of strata of the form S ∩ {t > 2 a } ∩ B n+1 r (resp. S ∩ {t > 2 a } ∩ S n r ). Such a stratum S is a product S×] -, "[ where , " > 0 are small and S is a stratum of X. A point (x 0 , t 0 ) is a critical point of t | S∩S n r if and only if x 0 ∈ S and rank        N 1 (x 0 ) 0 . . . . . . N cs (x 0 ) 0 x 0 t 0 0 1        < c S + 2,
where (N 1 (x 0 ), . . . , N c S (x 0 )) is a basis of the normal space to S at x 0 , that is if and only if rank

     N 1 (x 0 ) . . . N cs (x 0 ) x 0      < c S + 1.
But if x 0 is a point of S close to 0 but distinct from 0, the sphere S n-1 |x0| intersects the stratum S transversally. We conclude that the unique possible critical point of

t | S∩S n r is the point (0, r ). But v is in (-C 0 Y \ -C 0 Y ) ∩ S n-1 , so by Lemma 4.14, there is α > 0 such that ]0, α] • (-v) ⊂ T where T is a stratum of Y of dimension n. Hence {0}×]0, α] is included in T v . We conclude that for 0 < 1, (0, r ) is the only critical point of t |Zv∩{t> 2 a }∩B n+1 r
. Moreover it is a strict local maximum. Applying Theorem 3.1 in [START_REF] Dutertre | On the topology of semi-algebraic functions on closed semi-algebraic sets[END_REF] and Lemma 4.15, we get

χ Z v ∩ {t ≥ 2 a } ∩ B n+1 r -χ Z v ∩ {t = 2 a } ∩ B n+1 r = 0,
and

χ Z v ∩ {t ≥ 2 a } ∩ B n+1 r = ind(-t, Z v ∩ B n+1 r , (0, r )) = 1, because -t |Zv∩B n+1
r has a strict local minimum at the point (0, r ).

Corollary 4.17. Under the same assumptions, we have

χ (Lk(Z v ∩ {t ≥ 0})) = χ c Z v ∩ {t ≥ 0} ∩ {(x, t) | |x| = , t < 2 a } + 1.
Proof. Let h(x, t) be the semi-algebraic function defined by

h(x, t) = max |x|, a 2 t .
As explained by Durfee in [START_REF] Durfee | Neighborhoods of algebraic sets[END_REF], Section 3, the link of

Z v ∩ {t ≥ 0} is homeomorphic to Z v ∩ {t ≥ 0} ∩ {h = } for 0 < 1.
We have

Z v ∩ {t ≥ 0} ∩ {h = } = Z v ∩ {t ≥ 0} ∩ {|x| = , at 2 ≤ } Z v ∩ {t ≥ 0} ∩ {|x| ≤ , at 2 = }.
It is enough to use the previous lemma and the additivity of χ c .

4.5.

On the two sides of the kinematic formulas. We prove the existence of the left-hand sides of the kinematic formulas, and we show that both sides of the formulas are symmetric in X and Y . We also give a relation with the polar invariants.

Let (X, 0) ⊂ (R n , 0) and (Y, 0) ⊂ (R n , 0) be two germs of closed definable sets. We assume that X and Y are included in an open set U . Let 0 > 0 be such that

B n 0 ⊂ U . 1) Let us fix ( , δ) such that 0 ≤ ≤ 0 and 0 ≤ δ ≤ . Let A = (x, γ, v) ∈ R n × SO(n) × S n-1 | x ∈ X, x -δv ∈ γY, |x| ≤ .
It is a closed definable set. By Hardt's theorem applied to the projection π :

A → SO(n) × S n-1 , the function (γ, v) → χ (X ∩ (γY + δv) ∩ B n ) takes a finite number of values.
As in [START_REF] Bernig | Courbures intrinsèques dans les catégories analyticogéométriques[END_REF] we equip SO(n) with the Haar measure dγ, normalized in such a way that the volume of SO(n) is s n-1 . We equip S n-1 with the usual Riemanniann measure (or density) dv and SO(n) × S n-1 with the product measure dγdv. With this measure, the function (γ, v) → χ (X ∩ (γY + δv) ∩ B n ) is integrable and so the integral

SO(n)×S n-1 χ (X ∩ (γY + δv) ∩ B n ) dγdv
exists and is finite. Moreover the function

γ → S n-1 χ (X ∩ (γY + δv) ∩ B n ) dv is integrable and the function v → SO(n) χ (X ∩ (γY + δv) ∩ B n ) dγ
is integrable and we have

SO(n)×S n-1 χ (X ∩ (γY + δv) ∩ B n ) dγdv = SO(n) [ S n-1 χ (X ∩ (γY + δv) ∩ B n ) dv]dγ = S n-1 [ SO(n) χ (X ∩ (γY + δv) ∩ B n ) dγ]dv. Now let us fix > 0 such that 0 < ≤ 0 . By Hardt's theorem, for every (γ, v) ∈ SO(n) × S n-1 , there is a small interval ]0, δ [ such that the function δ → χ (X ∩ (γY + δv) ∩ B n ) is constant on ]0, δ [ and so lim δ→0 + χ (X ∩ (γY + δv) ∩ B n )
exists and is this constant value. Similarly as above, the function (γ, v, δ) → χ (X ∩ (γY + δv) ∩ B n ) takes a finite number of values and so, by Lebesgue's theorem, the function (γ, v) → lim δ→0 + χ (X ∩ (γY + δv) ∩ B n ) is integrable and we have lim

δ→0 + SO(n)×S n-1 χ (X ∩ (γY + δv) ∩ B n ) dγdv = SO(n)×S n-1 lim δ→0 + χ (X ∩ (γY + δv) ∩ B n ) dγdv. ( * ) Let us fix (γ, v) ∈ SO(n) × S n-1 and let B = (x, , δ) ∈ R n × {( , δ) | 0 < ≤ 0 , 0 < δ ≤ } | x ∈ X, x -δv ∈ Y, |x| ≤ .
It is a closed definable set. Applying the argument of the proof of Lemma 3.1, we see that there exists 0 < 1 ≤ 0 and a definable function r :]0, 1 ] → R continuous, monotone and strictly positive such that the function

( , δ ) → χ (X ∩ (γY + δv) ∩ B n ) is constant on {( , δ ) | 0 < < 1 , 0 < δ < r( )}. But we see that for ∈]0, 1 [ and 0 < δ < r( ), χ (X ∩ (γY + δ v) ∩ B n ) = lim δ→0 + χ (X ∩ (γY + δv) ∩ B n ) .
Therefore the limit lim →0 lim δ→0 + χ (X ∩ (γY + δv) ∩ B n ) exists and equals the above constant value. Always by Hardt's theorem, the function (γ, v, , δ) → χ (X ∩ (γY + δv) ∩ B n ) takes a finite number of values and so does the function (γ, v, ) → lim δ→0 + χ (X ∩ (γY + δv) ∩ B n ). By Lebesgue's theorem, the function

(γ, v) → lim →0 lim δ→0 + χ (X ∩ (γY + δv) ∩ B n ) is integrable and lim →0 SO(n)×S n-1 lim δ→0 + χ (X ∩ (γY + δv) ∩ B n ) dγdv = SO(n)×S n-1 lim →0 lim δ→0 + χ (X ∩ (γY + δv) ∩ B n ) dγdv.
Finally, by Equality ( * ) above, we have that lim

→0 [ lim δ→0 + SO(n)×S n-1 χ (X ∩ (γY + δv) ∩ B n ) dγdv]
exists and equals

SO(n)×S n-1 lim →0 lim δ→0 + χ (X ∩ (γY + δv) ∩ B n ) dγdv.
Definition 4.18. For two germs of closed definable sets (X, 0) ⊂ (R n , 0) and (Y, 0) ⊂ (R n , 0), we set

σ(X, Y, 0) = 1 s 2 n-1 SO(n)×S n-1 lim →0 lim δ→0 + χ (X ∩ (γY + δv) ∩ B n ) dγdv. 2) Let us fix such that 0 ≤ ≤ 0 . Let C = (x, γ, H) ∈ R n × SO(n) × G n-1 n | x ∈ X ∩ γY ∩ H, |x| = .
It is a closed definable set. By Hardt's theorem applied to the projection π :

C → SO(n)×G n n-1 , the function (γ, H) → χ X ∩ γY ∩ H ∩ S n
(γ, v) → G n-1 n χ (Lk(X ∩ γY ∩ H)) dH on SO(n) × S n-1 . Similarly it is easy to see that the function γ → χ (Lk(X ∩ γY )) is integrable on SO(n) and so is the function (γ, v) → χ (Lk(X ∩ γY )) on SO(n) × S n-1 .
3) By Lemmas 4.10 and 4.11, there exists a definable subset ∆ ⊂ SO(n

) × S n-1 of positive codimension such that for (γ, v) / ∈ ∆, lim →0 lim δ→0 + Λ 0 (X ∩ (γY + δv), X ∩ (γY + δv) ∩ B n ) = lim →0 lim δ→0 + χ (X ∩ (γY + δv) ∩ B n ) - 1 2 χ (Lk(X ∩ γY )) - 1 2s n-1 S n-1 χ (Lk(X ∩ γY ∩ {u * = 0})) du.
Therefore the function

(γ, v) → lim →0 lim δ→0 + Λ 0 (X ∩ (γY + δv), X ∩ (γY + δv) ∩ B n ) is integrable on SO(n) × S n-1 .
Definition 4.19. For two germs of closed definable set (X, 0) ⊂ (R n , 0) and (Y, 0) ⊂ (R n , 0), we set

Λ lim 0 (X, Y, 0) = 1 s 2 n-1 SO(n)×S n-1 lim →0 lim δ→0 + Λ 0 (X ∩ (γY + δv), X ∩ (γY + δv) ∩ B n ) dγdv.
We note that

Λ lim 0 (X, Y, 0) = σ(X, Y, 0) - 1 2s n-1 SO(n) χ (Lk(X ∩ γY )) dγ - 1 s 2 n-1 SO(n) S n-1 χ (Lk(X ∩ γY ∩ {u * = 0})) dudγ.
The two limits σ(X, Y, 0) and Λ lim 0 (X, Y, 0) are symmetric in X and Y , as explained in the next proposition. Proposition 4.20. For two germs of closed definable sets (X, 0) ⊂ (R n , 0) and

(Y, 0) ⊂ (R n , 0), we have σ(X, Y, 0) = σ(Y, X, 0) and Λ lim 0 (X, Y, 0) = Λ lim 0 (Y, X, 0). Proof. By Lemma 4.6, we know that lim →0 lim δ→0 + χ (X ∩ (γY + δv) ∩ B n ) = lim →0 lim δ→0 + χ ((X -δv) ∩ γY ∩ B n ) . The change of variables v → -v gives that S n-1 χ ((X -δv) ∩ γY ∩ B n ) dv = S n-1 χ ((X + δv) ∩ γY ∩ B n ) dv,
and so

S n-1 χ ((X -δv) ∩ γY ∩ B n ) dv = S n-1 χ γ -1 (X + δv) ∩ Y ∩ B n dv.
Hence, by the change of variables γ → γ -1 on SO(n), we are lead to compute

SO(n) [ S n-1 χ (γ(X + δv) ∩ Y ∩ B n ) dv]dγ.
But for γ ∈ SO(n), the change of variables u → γu gives that

S n-1 χ (γ(X + δu) ∩ Y ∩ B n ) du = S n-1 χ ((γX + δv) ∩ Y ∩ B n ) dv.
Finally we get that

SO(n) S n-1 χ (γ(X + δv) ∩ Y ∩ B n ) dvdγ = SO(n) S n-1 χ ((γX + δv) ∩ Y ∩ B n ) dvdγ.
It is enough to pass to the limits to get the equality σ(X, Y, 0) = σ(Y, X, 0). The result for Λ lim 0 (X, Y, 0) is obtained applying the relation between σ(X, Y, 0) and Λ lim 0 (X, Y, 0). Now let us relate σ(X, Y, 0) with the polar invariants of Comte and Merle [START_REF] Comte | Equisingularité réelle II : invariants locaux et conditions de régularité[END_REF].

Proposition 4.21. If H ∈ G n-k n , k ∈ {0, . . . , n}, then we have σ(X, H, 0) = σ k (X, 0).
Proof. The proof is straightforward for k = 0, because in this case σ(X,

R n , 0) = lim →0 χ(X ∩ B n ) = 1.
Let us assume that k > 0. First we note that

σ(X, H, 0) = 1 g n-k n G n-k n 1 s n-1 S n-1 lim →0 lim δ→0 + χ (X ∩ (H + δv) ∩ B n ) dv dH.
For H ∈ G n-k n , we recall that S k-1 H ⊥ is the unit sphere in H ⊥ and we denote by p H ⊥ the orthogonal projection onto

H ⊥ . If v ∈ S k-1 H ⊥ and w ∈ S n-1 are such that p H ⊥ (w) |p H ⊥ (w)| = v, then lim →0 lim δ→0 + χ (X ∩ (H + δv) ∩ B n ) = lim →0 lim δ→0 + χ (X ∩ (H + δw) ∩ B n ) . This implies that σ(X, H, 0) = 1 g n-k n G n-k n 1 s k-1 S k-1 H ⊥ lim →0 lim δ→0 + χ (X ∩ (H + δv) ∩ B n ) dv dH,
(see the proof of Corollary 5.7 in [START_REF] Dutertre | A Gauss-Bonnet formula for closed semi-algebraic sets[END_REF] for a similar argument or use the co-area formula). To end the proof, it is enough to show, with the notations of Section 3, that

N P i=1 χ P i • Θ(K p i , 0) = 1 s k-1 S k-1 P lim →0 lim δ→0 + χ X ∩ (P ⊥ + δv) ∩ B n dv. By Lemma 4.14, we know that if v ∈ C 0 K P i \ C 0 (K P i \ K P i ), there is δ > 0 such that ]0, δ] • v ⊂ K P i . Hence 1 s k-1 S k-1 P lim →0 lim δ→0 + χ X ∩ (P ⊥ + δv) ∩ B n dv = N P i=1 χ P i • vol(C 0 K P i ∩ S k-1 P ) s k-1
.

By [START_REF] Kurdyka | Densité des ensembles sous-analytiques[END_REF] Lemma 2.1,

vol(C0K P i ∩S k-1 P ) s k-1 is exactly Θ k (K P i , 0).
Remark 4.22. The equality

σ k (X, 0) = 1 g n-k n G n-k n 1 s k-1 S k-1 H ⊥ lim →0 lim δ→0 + χ (X ∩ (H + δv) ∩ B n ) dv dH,
is natural because the two sides of the equality coincide for they measure the same mean-value of Euler characteristics. It already appeared in [START_REF] Comte | Equisingularité réelle II : invariants locaux et conditions de régularité[END_REF] page 244 in the conic case, and we used it in [START_REF] Dutertre | Stratified critical points on the real Milnor fibre and integral-geometric formulas[END_REF][START_REF] Dutertre | Euler obstruction and Lipschitz-Killing curvatures[END_REF][START_REF] Dutertre | Lipschitz-Killing curvatures and polar images[END_REF] as a definition for the polar invariants. We prove it here for completeness.

We end this subsection with another symmetry result. Lemma 4.23. For two germs of closed definable sets (X, 0) ⊂ (R n , 0) and (Y, 0) ⊂ (R n , 0), we have

n i=0 Λ lim i (X, 0) • σ n-i (Y, 0) = n i=0 σ i (X, 0) • Λ lim n-i (Y, 0).
Proof. By Theorem 3.8, we get

n i=0 Λ lim i (X, 0) • σ n-i (Y, 0) = n-1 i=0 (σ i (X, 0) -σ i+1 (X, 0)) • σ n-i (Y, 0) +σ n (X, 0) • σ 0 (Y, 0) = n-1 i=0 σ i (X, 0) • σ n-i (Y, 0) - n-1 i=0 σ i+1 (X, 0) • σ n-i (Y, 0) + σ n (X, 0) • σ 0 (Y, 0).
Therefore we obtain

n i=0 Λ lim i (X, 0) • σ n-i (Y, 0) = σ 0 (X, 0) • σ n (Y, 0) + n-1 i=1 σ i (X, 0) • (σ n-i (Y, 0) -σ n-i+1 (Y, 0)) + σ n (X, 0) • (σ 0 (Y, 0) -σ 1 (Y, 0))
Another application of Theorem 3.8 gives the result.

A new spherical kinematic formula

We give a new spherical kinematic formula for two definable subsets of the unit sphere.

Let X ⊂ S n-1 be a compact definable set and let Y ⊂ S n-1 be a definable set, not necessarly compact. We recall that the Λi 's, i = 0, . . . , n -1, denote the spherical Lipschitz-Killing measures.

Proposition 5.1. The following kinematic formula holds:

1 s n-1 SO(n) χ c (X ∩ γY )dγ = n-1 i=0
Λi (X, X)

s i • 1 g i+1 n G i+1 n χ c (Y ∩ H)dH.
Proof. First step: We study the case Y compact. Applying the generalized spherical Gauss-Bonnet formula (see Theorem 1.2 in [START_REF] Bernig | Courbures intrinsèques dans les catégories analyticogéométriques[END_REF]) to χ(X ∩ γY ), we obtain

SO(n) χ(X ∩ γY )dγ = n-1 i=0,2,... 2 s i SO(n) Λi (X ∩ γY, X ∩ γY )dγ.
Then we apply the generalized spherical kinematic formula (see [START_REF] Bernig | Courbures intrinsèques dans les catégories analyticogéométriques[END_REF][START_REF] Fu | Kinematic formulas in integral geometry[END_REF]) to each Λi (X ∩ γY, X ∩ γY ) and we get

SO(n) χ(X ∩ γY )dγ = n-1 i=0,2,... 2 s i p+q=i+n-1 s i s n-1 s p s q Λp (X, X) Λq (Y, Y ).
Therefore we have

1 s n-1 SO(n) χ(X ∩ γY )dγ = n-1 i=0,2,... p+q=i+n-1 Λp (X, X) s p 2 Λq (Y, Y ) s q = p+q=n-1 Λp (X, X) s p 2 Λq (Y, Y ) s q + p+q=n+1 Λp (X, X) s p 2 Λq (Y, Y ) s q + • • • + p+q=e(n) Λp (X, X) s p 2 Λq (Y, Y ) s q ,
where e(n) = 2n -2 if n -1 is even or 2n -3 if n -1 is odd. This equality can be rewritten in the following way:

1 s n-1 SO(n) χ(X ∩ γY )dγ = Λn-1 (X, X) s n-1 q=0,2,... 2 Λq (Y, Y ) s q + Λn-2 (X, X) s n-2 q=1,3,... 2 Λq (Y, Y ) s q + • • • + Λ0 (X, X) 2 2 Λn-1 (Y, Y ) s n-1 .
In [START_REF] Dutertre | Euler characteristic and Lipschitz-Killing curvatures of closed semialgebraic sets[END_REF] pages 175-176, we proved that

1 g 1 n G 1 n χ(Y ∩ H)dH = 2 Λn-1 (Y, Y ) s n-1 , 1 g 2 n G 2 n χ(Y ∩ H)dH = 2 Λn-2 (Y, Y ) s n-2 ,
and for k ≥ 3,

1 g k-2 n G k-2 n χ(Y ∩ H)dH = i=2,4,... 2 Λn-k+i (Y, Y ) s n-k+i = q=n-k+2,n-k+4,... 2 Λq (Y, Y ) s q .
Applying these relations, we get that

1 s n-1 SO(n) χ(X∩γY )dγ = Λn-1 (X, X) s n-1 χ(Y )+ Λn-2 (X, X) s n-2 1 g n-1 n G n-1 n χ(Y ∩H)dH + Λn-3 (X, X) s n-3 1 g n-2 n G n-2 n χ(Y ∩ H)dH + • • • + Λ0 (X, X) s 0 1 g 1 n G 1 n χ(Y ∩ H)dH.
Second step: Let Y ⊂ S n-1 be compact and let K Y be a compact definable set. By the first step, we have

1 s n-1 SO(n) χ(X ∩ γY )dγ = n-1 i=0 Λ(X, X) s i • 1 g i+1 n G i+1 n χ(Y ∩ H)dH, and
1 s n-1 SO(n) χ(X ∩ γK)dγ = n-1 i=0 Λ(X, X) s i • 1 g i+1 n G i+1 n χ(K ∩ H)dH. For each γ ∈ SO(n), γY = (γY \ γK) γK = γ(Y \ K) γK because γ is bijective. Hence χ(X ∩ γY ) = χ c (X ∩ γ(Y \ K)) + χ(X ∩ γK) and 1 s n-1 SO(n) χ c (X ∩ γ(Y \ K)) dγ = n-1 i=0
Λi (X, X)

s i • 1 g i+1 n G i+1 n [χ(Y ∩ H) -χ(K ∩ H)] dH = n-1 i=0
Λi (X, X)

s i • 1 g i+1 n G i+1 n χ c ((Y \ K) ∩ H) dH.
This gives the result for the set Y \ K. Third step: We prove the general case. Since Y is definable, it admits the following cell decomposition Y = r j=1 C j , where each C j is a definable subset homeomorphic to a unit cube ]0, 1[ dj . By the second step, the result is valid for each cell C j , because C j and C j \ C j are compact and definable. By additivity of χ c , we have 1

s n-1 SO(n) χ c (X ∩ γY )dγ = 1 s n-1 SO(n) χ c X ∩ γ( r j=1 C j ) dγ = 1 s n-1 SO(n) χ c X ∩ ( r j=1 γC j ) dγ = 1 s n-1 SO(n) χ c r j=1 (X ∩ γC j ) dγ = r j=1 1 s n-1 SO(n) χ c (X ∩ γC j )dγ.
Applying the second step, we obtain

1 s n-1 SO(n) χ c (X ∩ γY )dγ = r j=1 n-1 i=0
Λi (X, X)

s i • 1 g i+1 n G i+1 n χ c (C j ∩ H)dH = n-1 i=0
Λi (X, X)

s i • 1 g i+1 n G i+1 n r j=1 χ c (C j ∩ H)dH = n-1 i=0
Λi (X, X)

s i • 1 g i+1 n G i+1 n χ c (Y ∩ H)dH.

A second kinematic formula in the unit ball

We deduce from the previous spherical kinematic a new kinematic formula for definable subsets of the unit ball.

Let X ⊂ R n be a closed conic definable set. Let Y ⊂ B n be another definable set.

Proposition 6.1. The following kinematic formula holds:

1 s n-1 SO(n) χ c (X ∩ γY )dγ = n i=0 Λ i (X, X ∩ B n ) b i • 1 g i n G i n χ c (Y ∩ H)dH.
Proof. Let us assume first that 0 / ∈ Y and let φ be the following definable mapping:

φ : B n \ {0} → S n-1 x → x |x|
. By Hardt's theorem, there exists a definable partition of φ(Y ), φ(Y ) = r j=1 W j , such that for j ∈ {1, . . . , r}, the mapping φ |Y ∩φ -1 (Wj ) : φ -1 (W j )∩Y → W j is trivial. By additivity and multiplicity of χ c , we can write χ c (Y ) = r j=1 α j χ c (W j ), where α j = χ c (F j ) with F j the fibre of φ |Y ∩φ -1 (Wj ) . Let us set X * = (X ∩ B n ) \ {0}.

If w belongs to φ(Y ) ∩ φ(X * ) then w = φ(y) = φ(x) with y ∈ Y and x ∈ X * . Since X is conic, y belongs to X * and so φ(Y ∩ X

* ) = φ(Y ) ∩ φ(X * ). Therefore φ(Y ∩ X * ) = r j=1 W j ∩ φ(X * ). Note that if w ∈ φ(X * ) then φ -1 (w) ⊂ X * by the conic structure of X. Hence if w j ∈ W j ∩ φ(X * ), φ -1 (w j ) ∩ Y ∩ X = φ -1 (w j ) ∩ Y
and χ c (φ -1 (w j )∩Y ∩X) = α j . Applying again Hardt's theorem to each W j ∩φ(X * ) if necessary, one can conclude as above that

χ c (Y ∩ X) = r j=1 α j χ c (W j ∩ φ(X * )).
Let γ ∈ SO(n). Since φ • γ = γ • φ and γ is a definable homeomorphism, φ(γY ) = r j=1 γW j and for j ∈ {1, . . . , r}, the mapping φ |γY ∩φ -1 (γWj ) : γY ∩ φ -1 (γW j ) → γW j is trivial, with fibre homeomorphic to F j . As above, we can write

χ c (X ∩ γY ) = r j=1 α j χ c (γW j ∩ φ(X * )) .
We can apply Proposition 5.1 to the sets W j and φ(X * ). We get

1 s n-1 SO(n) χ c (X ∩ γY ) dγ = r j=1 α j 1 s n-1 SO(n) χ c (γW j ∩ φ(X * ))dγ = r j=1 α j n-1 i=1
Λi (φ(X * ), φ(X * ))

s i • 1 g i+1 n G i+1 n χ c (W j ∩ H)dH = n-1 i=1
Λi (φ(X * ), φ(X * ))

s i • 1 g i+1 n G i+1 n r j=1 α j χ c (W j ∩ H)dH.
Since H is conic, r j=1 α j χ c (W j ∩H) = χ c (Y ∩H) by the above argument. Applying Corollary 3.5 in [START_REF] Dutertre | Euler characteristic and Lipschitz-Killing curvatures of closed semialgebraic sets[END_REF], we obtain the following equality:

1 s n-1 SO(n) χ c (X ∩ γY )dγ = n-1 i=0 Λ i+1 (X, X ∩ B n ) b i+1 • 1 g i+1 n G i+1 n χ c (Y ∩ H)dH = n i=1 Λ i (X, X ∩ B n ) b i • 1 g i n G i n χ c (Y ∩ H)dH,
which is the expected one when 0

/ ∈ Y . If 0 ∈ Y then χ c (X ∩ γY ) = χ c (X ∩ γY * ) + 1 and χ c (Y ∩ H) = χ c (Y * ∩ H) + 1 where Y * = Y \ {0}. Therefore we have 1 s n-1 SO(n) χ c (X ∩ γY )dγ = 1 + 1 s n-1 SO(n) χ c (X ∩ γY * )dγ = 1 + n i=1 Λ i (X, X ∩ B n ) b i • 1 g i n G i n χ c (Y * ∩ H)dH = 1 + n i=1 Λ i (X, X ∩ B n ) b i • 1 g i n G i n χ c (Y ∩ H)dH - n i=1 Λ i (X, X ∩ B n ) b i .
But by Corollary 5.2 in [START_REF] Dutertre | Euler characteristic and Lipschitz-Killing curvatures of closed semialgebraic sets[END_REF], we know that 1 -

n i=1 Λi(X,X∩B n ) bi = Λ 0 (X, X ∩ B n ).

A infinitesimal kinematic formula for conic sets

We prove a first version of the infinitesimal principal kinematic formula for closed conic definable sets.

Let X, Y ⊂ R n be two closed conic definable sets. We keep the notations used in Section 4.

Lemma 7.1. There exists δ 1 > 0 such that for 0 < δ ≤ δ 1 , the topological types of Z v,δ ∩ B n does not depend on the choice δ. Moreover, we have

lim δ→0 χ(Z v,δ ∩ B n ) = lim →0 lim δ→0 χ(Z v,δ ∩ B n ).
Proof. By Lemma 4.4, we know that there exists 0 > 0 such that for 0 < ≤ 0 , there exits δ such that for 0 < δ ≤ δ , the topological type of Z v,δ ∩ B n does not depend on the choice of the couple ( , δ). Let us fix such a couple ( , δ). Let θ : R n → R n be the diffeomorphism θ

(x) = 1 x. Then θ (Z v,δ ∩ B n ) = Z v, δ ∩ B n .
Since lim δ→0 δ = 0, we get the result.

We are in position to state a first infinitesimal kinematic formula in the conic setting.

Proposition 7.2. Let X, Y ⊂ R n be two closed conic definable sets. The following kinematic formula holds:

1 s 2 n-1 SO(n)×S n-1 lim δ→0 + χ (X ∩ (γY + δv) ∩ B n ) dγdv = n i=0 Λ i (X, X ∩ B n ) b i • σ n-i (Y, 0).
Proof. Let us fix δ > 0. By the change of variable u = γv, we have that for γ ∈ SO(n)

S n-1 χ (X ∩ (γY + δu) ∩ B n ) du = S n-1 χ (X ∩ γ(Y + δv) ∩ B n ) dv.
Applying Proposition 6.1 to X ∩ B n and (Y + δv) ∩ B n , we get that 1

s 2 n-1 SO(n)×S n-1 χ (X ∩ γ(Y + δv) ∩ B n ) dγdv = 1 s 2 n-1 S n-1 SO(n) χ (X ∩ γ(Y + δv) ∩ B n ) dγdv = n i=1 Λ i (X, X ∩ B n ) b i • 1 g i n s n-1 S n-1 G i n χ ((Y + δv) ∩ B n ∩ H) dHdv,
and so that 1

s 2 n-1 SO(n)×S n-1 χ (X ∩ γ(Y + δv) ∩ B n ) dvdγ = n i=1 Λ i (X, X ∩ B n ) b i • 1 g i n G i n 1 s n-1 S n-1 χ ((Y + δv) ∩ B n ∩ H) dvdH.
Passing to the limit as δ → 0 + and using Lebesgue's theorem, we obtain that 1

s 2 n-1 SO(n)×S n-1 lim δ→0 + χ (X ∩ (γY + δv) ∩ B n ) dvdγ If dimY ≤ n -1 then we can put ∆ Y = S n-1 ∩ (-Y ).
If dimY = n then let Y be the union of the strata of Y of dimension less than or equal to n -1. If v ∈ (-Y ) \ (-Y ), then by Corollary 4.17, we have

χ Lk( CX u ∩ Y v ∩ {t ≥ 0}) = 1 + χ c CX u ∩ Y v ∩ {t ≥ 0} ∩ {ω = , t < 2 a } , χ Lk( X ∩ Y v ∩ {t ≥ 0}) = 1 + χ c X ∩ Y v ∩ {t ≥ 0} ∩ {ω = , t < 2 a } , for 0 < 1 and where a is such that Y v ∩ {t ≥ 0} ⊂ {(x, t) | ω(x)
≥ at} in a neighborhood of (0, 0).

As above let us choose 1 ≥ 0 such that for 0

< u ≤ 1 , χ Lk( X ∩ Y v ∩ {t ≥ 0}) = 1 + χ c X ∩ Y v ∩ {t ≥ 0} ∩ {ω = u, t < 2 a u} , and
χ c X ∩ Y v ∩ {t ≥ 0} ∩ {ω = u, t < 2 a u} = χ c CX u ∩ Y v ∩ {t ≥ 0} ∩ {ω = u, t < 2 a u} .
Using the same homeomorphism as above, we can conclude that

χ Lk( CX u ∩ Y v ∩ {t ≥ 0}) = 1 + χ c CX u ∩ Y v ∩ {t ≥ 0} ∩ {ω = u, t < 2 a u} . So if dimY = n, we put ∆ Y = S n-1 ∩ (-Y ).
Proposition 8.2. If Y ⊂ R n is a closed conic definable set, then for any germ of closed definable set (X, 0) ⊂ (R n , 0), the following principal kinematic formula holds:

σ(X, Y, 0) = n i=0 Λ lim i (X, Y, 0) • σ n-i (Y, 0).
Proof. By Lemma 4.10, there exists a definable subset Σ X,Y ⊂ SO(n) of positive codimension such that for γ / ∈ Σ X,Y , X and γY satisfy Condition (1). Let us fix γ / ∈ Σ X,Y . By Lemma 4.11, there exists a definable subset Γ X,γY ⊂ S n-1 of positive codimension such that for v / ∈ Γ X,γY , v satisfies Condition (2). Let us choose v / ∈ Γ X,γY . By Lemma 4.2, the function t : X ∩ (γY ) v → R has an isolated stratified critical point at (0, 0). Applying Lemma 3.1 in [START_REF] Dutertre | On the topology of non-isolated real singularities[END_REF] and Lemma 4.4, we obtain that lim

→0 lim δ→0 + χ (X ∩ (γY + δv) ∩ B n ) = χ Lk( X ∩ (γY ) v ∩ {t ≥ 0}) , and
σ(X, Y, 0) = 1 s 2 n-1 SO(n)×S n-1 χ Lk( X ∩ (γY ) v ∩ {t ≥ 0}) dγdv.
Of course the same equality is true if we replace X with CX u . By Proposition 7.2 for 0 < u ≤ 0 , we have

σ(CX u , Y, 0) = n i=0 Λ i (CX u , CX u ∩ B n ) b i • σ n-i (Y, 0).
Proof. The partition ∪ s i=0 S i ×]0, 0 [ gives a Whitney stratification of X×]0, 0 [. Let φ be the diffeomorphism φ(x, u) = ( 1 u x, u). We have Γ Si = φ(S i ×]0, 0 [) for i = 0, . . . , s and Γ X = φ(X×]0, 0 [). This gives the result for Whitney's conditions are C 1 -invariant.

We can equip C 0 X with a definable stratification (Σ k ) l k=0 where Σ 0 = {0} and Σ k is conic. This is possible for example by considering a Whitney stratification of C 0 X ∩ S n-1 and extending it to C 0 X using the conic structure. Lemma 8.9. Let Γ S be a stratum of Γ X and let Σ be a stratum of C 0 X such that Σ ⊂ Γ S \ Γ S . The set of points x in Σ such that the Thom (a u )-condition is not satisfied at x for the pair (Γ S , Σ) is a conic definable set of positive codimension.

Proof. By [START_REF] Bekka | Regular stratification of subanalytic sets[END_REF] and [START_REF] Loi | Thom stratifications for functions definable in o-minimal structures on (R, +[END_REF], we already know that this set is definable of positive codimension in Σ. If x is in this set, then there exists a sequence of points p n = (y n , u n ) in Γ S such that p n → (x, 0) and T x Σ lim n→+∞ T pn (Γ S ∩ {u = u n }). Let λ > 0, by the conic structure of Σ, T λx Σ = T x Σ. Moreover q n = (λy n , un λ ) ∈ Γ S and

T pn (Γ S ∩ {u = u n }) = T qn (Γ S ∩ {u = un λ }).
Since the Thom (a u )-condition is stratifying and taking a refinement if necessary, we can assume that the Thom (a u )-condition is satisfied for any pair of strata (Γ S , Σ) (see [START_REF] Lojasiewicz | Stratifications sous-analytiques. Condition de Verdier[END_REF] for the argument). This induces a Whitney stratification of Γ

X compatible with C 0 X × {0}. Namely if S ⊂ R n+1 is a stratum Γ X then S = {(x, t, u) | (x, u) ∈ S } is a stratum of Γ X .
Lemma 8.10. This induced stratification of Γ X satisfies the Thom (a u )-condition.

Proof. Let (x n , t n , u n ) n∈N be a sequence of points in Γ X that tends to (x, t, 0). We can assume that (x n , t n , u n ) lies in a stratum S 1 = {(x , t , u ) | (x , u ) ∈ S 1 } and that (x, t, 0) ∈ S 0 , where S 0 = {(x , t , 0) | (x , 0) ∈ S 0 }. Since the pair (S 1 , S 0 ) satisfies the Thom (a u )-condition, T (x,0) S 0 ⊂ lim n→+∞ T (xn,un) (S 1 ∩ {u = u n }). But

T (xn,tn,un) S 1 = (ν, τ, ξ) | (ν, ξ) ∈ T (xn,un) S 1 , and
T (x,t,0) S 0 = (ν, τ, 0) | (ν, 0) ∈ T (x,0) S 0 .
It is straightforward to conclude using the fact that T

(xn,un) (S 1 ∩ {u = u n }) = (T (xn,un) S 1 ) ∩ {u = 0} and T (xn,tn,un) ( S 1 ∩ {u = u n }) = (T (xn,tn,un) S 1 ) ∩ {u = 0} if n is sufficiently big.
Similarly we can equip Γ Y with a Whitney definable stratification compatible with C 0 Y × {0}, that satisfies the Thom (a u )-condition and such that the strata of C 0 Y are conic. This induces a Whitney stratification of (Γ

Y ) v compatible with (C 0 Y ) v × {0}. Namely if T ⊂ R n+1 is a stratum of Γ Y then T v = {(y, t, u) | (y - tv, u) ∈ T } is a stratum of (Γ Y ) v (see Section 4).
Lemma 8.11. This induced stratification of (Γ Y ) v satisfies the Thom (a u )-condition. Assume that the above result is not true. Then we can find a sequence of positive real numbers (u n ) n∈N that tends to 0 and a sequence of points (

x n , t n ) n∈N in CX un ∩ Y v ∩ {t > 0} ∩ {0 < ω ≤ u n } ∩ B n+1 such that CX un ∩ {ω = ω(x n )} and Y v do not intersect transversally at (x n , t n ).
We can assume that the sequence (x n , t n ) n∈N is included in a unique stratum T v , where T is a stratum of Y . Moreover we can assume that there is a stratum S = {0} of X such that for each n ∈ N, (x n , t n ) ∈ CS un . Since tn ω(xn) ≤ The points p n := ( 1 un w n , tn ω(xn) , u n ) belong to the stratum Γ S of Γ X and the point (x, t) belongs to a stratum Σ of C 0 X. By the Thom (a u )-condition, we have

T (x,t) Σ ⊂ lim n→+∞ T pn ( Γ S ∩ {u = u n }).
Since Σ is Σ×R and Σ is conic, Σ intersects {ω = 1} transversally. By the Thom (a u )-condition, {ω = 1} intersects Γ S ∩ {u = u n } transversally for n big enough, and so The points q n := ( xn ω(xn) , tn ω(xn) , ω(x n )) are in the stratum (Γ T ) v of (Γ Y ) v and the point (x, t) is in a stratum Σ v of (C 0 Y ) v . By the Thom (a u )-condition, we have This contradicts the construction of the sequence (x n , t n ) and ends the proof of this auxiliary lemma.

T (x,t) ( Σ ∩ {ω = 1}) ⊂ lim n→+∞ T pn ( Γ S ∩ {ω = 1} ∩ {u = u n }). But T pn ( Γ S ∩ {ω = 1} ∩ {u = u n }) = T (wn, tn ω(xn) ) ( S ∩ {ω = u n })
T (x,t) Σ v ⊂ lim n→+∞ T qn ( (Γ T ) v ∩ {u = ω(x n )}). But T qn ( (Γ T ) v ∩ {u = ω(x n )}) = T (xn,tn) T v and so T (x,t) Σ v ⊂ lim n→+∞ T (xn,tn) T v . Since v / ∈ Γ C0X∩{ω=1},
Similarly the following second auxiliary lemma holds.

Lemma 8.14. There exists 0 < u 2 ≤ u 0 such that for 0 < u ≤ u 2 and for x ∈ CX u ∩ Y ∩ {0 < ω ≤ u} ∩ B n v , the sets CX u ∩ {ω = ω(x)} and Y intersect transversally (in the stratified sense) in R n at x.

Let us choose u > 0 such that u ≤ min{ 1 , u 1 , u 2 }, where 1 is such that for 0

< u ≤ 1 , χ Lk( X ∩ Y v ∩ {t ≥ 0}) = χ X ∩ Y v ∩ {t ≥ 0} ∩ {ω = u} .
Then for (x, t) ∈ CX u ∩ Y v ∩{0 < ω ≤ u}∩{t > 0}∩B n+1 v , CX u ∩{ω = ω(x)}∩{t > 0} and Y v ∩ {t > 0} intersect transversally (in the stratified sense) at (x, t). This implies that CX u ∩{t > 0} and Y v ∩{t > 0} intersect transversally at (x, t) and that {ω = ω(x)} intersects CX u ∩ Y v ∩ {t > 0} transversally at (x, t), and so (x, t) is not a stratified critical point of ω | CXu∩ Yv∩{t>0} . Similarly if (x, 0)

∈ CX u ∩ Y v ∩ {0 < ω ≤ u}∩B
χ Lk( CX u ∩ Y v ∩ {t ≥ 0}) = 1 + χ c CX u ∩ Y v ∩ {t ≥ 0} ∩ {ω = , t < 2 a } , χ Lk( X ∩ Y v ∩ {t ≥ 0}) = 1 + χ c X ∩ Y v ∩ {t ≥ 0} ∩ {ω = , t < 2 a } ,
for 0 < 1 and where a > 0 is such that Y v ∩ {t ≥ 0} ⊂ {(x, t) | ω(x) ≥ at} in a neighborhood of (0, 0).

As above let us choose 1 > 0 such that for 0

< u ≤ 1 , χ Lk( X ∩ Y v ∩ {t ≥ 0}) = 1 + χ c X ∩ Y v ∩ {t ≥ 0} ∩ {ω = u, t < 2 a u} .
Since

χ c X ∩ Y v ∩ {t ≥ 0} ∩ {ω = u, t < 2 a u} = χ c CX u ∩ Y v ∩ {t ≥ 0} ∩ {ω = u, t < 2 a u} ,
we have to prove that

χ Lk( CX u ∩ Y v ∩ {t ≥ 0}) = 1 + χ c CX u ∩ Y v ∩ {t ≥ 0} ∩ {ω = u, t < 2 a u} ,
if u is small enough. By the previous case, we know that for u small enough and for v / ∈ Γ C0X∩{ω=1},C0Y , Proof. By Lemma 4.10, there exists a definable subset Σ X,Y ⊂ SO(n) of positive codimension such that for γ / ∈ Σ X,Y , X and γY satisfy Condition [START_REF] Bekka | Regular stratification of subanalytic sets[END_REF]. Let us fix γ / ∈ Σ X,Y . By Lemma 4.11, there exists a definable subset Γ X,γY ⊂ S n-1 of positive codimension such that for v / ∈ Γ X,γY , v satisfies Condition (2). Let us choose v / ∈ Γ X,γY . By Lemma 4.2, the function t : X ∩ (γY ) v → R has an isolated stratified critical point at (0, 0). Applying Lemma 3.1 in [START_REF] Dutertre | On the topology of non-isolated real singularities[END_REF] and Lemma 4. Λi (Lk(X), Lk(X))

ω : CX u ∩ Y v ∩ {0 < ω ≤ u} ∩ {t ≥ 0} ∩ B n+1 → R
s i • 1 g i+1 n G i+1 n χ (Lk(Y ∩ H)) dH,
which can be rewritten in the following way: 1 s n-1 SO(n) χ (Lk(X ∩ γY )) dγ

Proof. When X and Y are conic, it enough to apply the spherical kinematic formulas. The general case can be deduced as we have already done in several previous proofs.

Corollary 9.2. Let (X, 0) ⊂ (R n , 0) and (Y, 0) ⊂ (R n , 0) be two germs of closed definable sets. The following principal kinematic formula holds:

SO(n) lim →0 Λ 0 (X ∩ γY, X ∩ γY ∩ B n )dγ = n i=0 Λ lim i (X, 0) •   n-i j=0 Λ lim j (Y, 0)   .
Proof. Apply Proposition 9.1 and Corollary 3.6.

Proof.

  Let d : R n → R be the distance function to B. It is a continuous definable function on an open definable neighborhood O of B. Let

  [a, b] = {z | z = λa + (1 -λ)b, λ ∈ [0, 1]} . Since S is open, there exists 0 < η ≤ 1 such that [a, a + η(b -a)[ is included in S. Let η a be the supremum of such η's. The point a + η a (b -a) lies in S \ S. Actually if a + η a (b -a) is in S then there exists η > η a such that a + η (b -a) ∈ S, because S is open. We conclude that the segment [a, b] intersects S \ S.

  -1 takes a finite number of values. As above, we deduce that the function (γ, H) → χ (Lk(X ∩ γY ∩ H)) takes a finite number of values. We equip G n-1 n with the SO(n)-invariant measure (or density) dH and SO(n) × G n-1 n with the product measure. With this measure, the function (γ, H) → χ (Lk(X ∩ γY ∩ H)) is integrable and so the integral SO(n)×G n-1 n χ (Lk(X ∩ γY ∩ H)) dγdH exists. Moreover the function γ → G n-1 n χ (Lk(X ∩ γY ∩ H)) dH is integrable and so is the function

  and so T (x,t) ( Σ ∩ {ω = 1}) ⊂ lim n→+∞ T (wn, tn ω(xn ) ) ( S ∩ {ω = u n }). We note that T (wn, tn ω(xn ) ) ( S ∩ {ω = u n }) = T (xn,tn) ( CS un ∩ {ω = ω(x n )}), by the conic structure of CS un .

  C0Y , C 0 X ∩ {ω = 1} and (C 0 Y ) v intersect transversally (in the stratified sense). But C 0 X ∩ {ω = 1} = C 0 X ∩ {ω = 1}, and we conclude that T (x,t) ( Σ ∩ {ω = 1}) + T (x,t) Σ v = R n+1 . Therefore lim n→+∞ T (xn,tn) ( CS un ∩ {ω = ω(x n )}) + lim n→+∞ T (xn,tn) T v = R n+1 ,and so, for n big enoughT (xn,tn) ( CS un ∩ {ω = ω(x n )}) + T (xn,tn) T v = R n+1 .

  n+1

v

  , then (x, 0) is not a stratified critical point of ω | CXu∩ Yv∩{t=0} . Hence we conclude that ω :CX u ∩ Y v ∩ {0 < ω ≤ u} ∩ {t ≥ 0} ∩ B n+1 v → R is a stratified submersion and so that χ Lk( CX u ∩ Y v ∩ {t ≥ 0}) = χ CX u ∩ Y v ∩ {t ≥ 0} ∩ {ω = u} . Therefore if dimC 0 Y ≤ n -1, we can take ∆ X,Y = (-C 0 Y ) ∩ S n-1 ∪ Γ C0X∩{ω=1},C0Y .If dimC 0 Y = n then dimY = n. Let Y be the union of the strata of Y of dimension less than or equal to n -1. If v ∈ (-C 0 Y ) \ (-C 0 Y ), we know that

  is a stratified submersion, for an appropriate > 0. Since the strata ofY v \ Y v have dimension n + 1 and the strata of Y \ Y have dimension n, ω : CX u ∩ Y v ∩ {0 < ω ≤ u} ∩ {t ≥ 0} ∩ B n+1 → Ris a stratified submersion by the conic structure of CX u . For the same reason and becauseY v ∩ {t ≥ 0} ∩ {t = 2 a ω(x)} = {(0, 0)}, we see that ω : CX u ∩ Y v ∩ {0 < ω ≤ u} ∩ {t = 2 a ω(x)} → R is a stratified submersion. Hence for 0 < ≤ u, χ c CX u ∩ Y v ∩ {t ≥ 0} ∩ {ω = u, t < 2 a u} = χ c CX u ∩ Y v ∩ {t ≥ 0} ∩ {ω = , t < 2 a } . If dimC 0 Y = n, we take ∆ X,Y = (-C 0 Y ) ∩ S n-1 ∪ Γ C0X∩{ω=1},C0Y .Theorem 8.15. Let (X, 0) ⊂ (R n , 0) and (Y, 0) ⊂ (R n , 0) be two germs of closed definable sets. The following principal kinematic formula holds:σ(X, Y, 0) = n i=0 Λ lim i (X, 0) • σ n-i (Y, 0).

1 χχs 2 n

 12 ∩ (γY + δv) ∩ B n ) = χ Lk( X ∩ (γY ) v ∩ {t ≥ 0}) , Lk( X ∩ (γY ) v ∩ {t ≥ 0}) dvdγ.Of course the same equality is true if we replace X with CX u . By Corollary 8.3, for 0 < u ≤ 0 , we haveσ(CX u , Y, 0) = n i=0 Λ i (CX u , CX u ∩ B n ) b i • σ n-i (Y, 0). Since C 0 (γY ) = γ(C 0 Y ),by Lemma 4.10 there exists a definable subset Σ C0X,C0Y ⊂ SO(n) of positive codimension such that for γ / ∈ Σ C0X,C0Y , C 0 X and C 0 (γY ) satisfy Condition (1). For γ / ∈ Σ C0X,C0Y and v / ∈ ∆ X,γY , lim u→0Lk( CX u ∩ (γY ) v ∩ {t ≥ 0}) = χ Lk( X ∩ (γY ) v ∩ {t ≥ 0}) .Hence, by Hardt's theorem and Lebesgue's theorem,lim u→0 σ(CX u , Y, 0) = σ(X, Y, 0).We end the proof as in Proposition 8.2.Let us specify this kinematic formula when d + e = n, d = dimX and e = dimY . We denote by X d (resp. Y e ) the union of the top-dimensional strata of X (resp. Y ).Corollary 8.16. Let (X, 0) ⊂ (R n , 0) and (Y, 0) ⊂ (R n , 0) be two germs of closed definable sets such that d + e = n, where d = dimX and e = dimY .The following formula holds:1 s 2 n-1 SO(n)×S n-1 lim →0 lim δ→0 + # X d ∩ (γY e + δv) ∩ B n dγdv = Θ d (X) • Θ e (Y ). Proof. For γ generic in SO(n) and v generic in S n-1 , lim →0 lim δ→0 + χ (X ∩ (γY + δv) ∩ B n ) = lim →0 lim δ→0 + # X d ∩ (γY e + δv) ∩ B n .Let us formulate now the second principal kinematic formula. Theorem 8.17. Let (X, 0) ⊂ (R n , 0) and (Y, 0) ⊂ (R n , 0) be two germs of closed definable sets. The following principal kinematic formula holds:Λ lim 0 (X, Y, 0) = n i=0 Λ lim i (X, 0) • Λ lim n-i (Y, 0). Proof. Let us compute the integrals 1 s n-1 SO(n) χ (Lk(X ∩ γY )) dγ,and1 -1 SO(n) S n-1 χ (Lk(X ∩ γY ∩ {u * = 0})) dudγ.Let us assume first that X and Y are conic closed definable sets. We have already computed the first integral in the proof of Proposition 5.1 and we have found that 1 s n-1 SO(n) χ (Lk(X ∩ γY )) dγ =

  , then w n ∈ CS un ∩ {ω = u n } and so w n ∈ S ⊂ X. Therefore (w n , tn ω(xn) ) belongs to X and ( 1 un w n , tn ω(xn) ) tends to (x, t). By Lemma 8.4 and Corollary 8.5, this implies that (x, t) is in C 0 X. Moreover since ω( wn un ) = 1, (x, t) belongs to C 0 X ∩ {ω = 1}. On the other hand, (x n , t n ) ∈ Y v and so by Lemma 8.6 and Corollary 8.7, (x, t) belongs to (C 0 Y ) v .

		1 a , taking
	a subsequence if necessary, we can assume that ( xn ω(xn) , tn ω(xn) ) tends to (x, t). Let
	w n = u n	xn ω(xn)

ANR project LISA 17-CE400023-01.

By Lemmas 4.6 and 7.1 applied to Y and H, we have that lim

and so

), by Proposition 4.21.

The principal kinematic formulas

We prove our main results : the principal kinematic formulas for germs of closed definable sets. We will use the kinematic formula for closed conic definable sets proved in the previous section. We will proceed in several steps.

We keep the notations used in Section 4. For convenience, we also use the notation ω(x) for |x|, if x is in R n .

First step: (X, 0) ⊂ (R n , 0) is a germ of closed definable set, Y ⊂ R n is a closed conic definable set.

We assume that X is included in an open neighborhood U of 0. Let 0 > 0 be such that B n 0 ⊂ U . For 0 < u ≤ 0 , we set X u = X ∩ S n-1 u and we denote by CX u the cone over X u , i.e.:

, then by Corollary 4.13, we have

For any > 0, the mapping

Hence, by Hardt's theorem and Lebesgue's theorem, lim u→0 σ(CX u , Y, 0) = σ(X, Y, 0).

Moreover using Proposition 3.6 in [START_REF] Dutertre | Euler characteristic and Lipschitz-Killing curvatures of closed semialgebraic sets[END_REF], for i ∈ {1, . . . , n -2}, we have

, and lim u→0 χ(CX u ∩ S n-1 ∩ H) = χ (Lk(X ∩ H)). Passing to the limit as u → 0 and applying Theorem 5.1 in [START_REF] Dutertre | Euler characteristic and Lipschitz-Killing curvatures of closed semialgebraic sets[END_REF], we get that

The same proof works for i = n -1 and i = n. Combining all these equalities, we get the result.

Corollary 8.3. Let X ⊂ R n be a closed conic definable set. For any germ of closed definable set (Y, 0) ⊂ (R n , 0), the following principal kinematic formula holds:

Proof. By Proposition 4.20, we know that σ(X, Y, 0) = σ(Y, X, 0), and by Lemma 4.23 that

). Then we apply the previous proposition.

Third step: (X, 0) ⊂ (R n , 0) and (Y, 0) ⊂ (R n , 0) are germs of closed definable sets.

We assume that X and Y are included in an open neighborhood U of 0. Let

We recall that the tangent cones of X and

We will now define two tangent cones associated with X and Y v and will relate them to C 0 X and

The following lemma is easy to prove.

Lemma 8.4. A point (x, t) belongs to Γ X ∩ (R n+1 × {0}) if and only if there is a sequence of points (x n , t n ) n∈N in X and a sequence of positive real numbers (u n ) n∈N such that u n → 0 and ( xn un , t n ) → (x, t).

Corollary 8.5. We have C 0 X = Γ X ∩ (R n+1 × {0}).

Proof. If (x, t) ∈ C 0 X then there is a sequence of points (x n ) n∈N in X and a sequence of positive real numbers (u n ) n∈N such that u n → 0 and xn un → x. Applying the previous lemma to the sequences (x n , t) and (u n ), we see that

), then there is a sequence of points (x n t n ) n∈N in X × R and a sequence of positive real numbers (u n ) n∈N such that u n → 0 and ( xn un , t n ) → (x, t). This implies that x ∈ C 0 X and so that (x, t) ∈ C 0 X.

Proof. If (y, t) ∈ (C 0 Y ) v then there is a sequence of points (y n ) n∈N in Y and a sequence of positive real numbers (u n ) n∈N such that u n → 0 and yn un → y -tv. For n ∈ N, (y n + u n tv, u n tv) is in Y v and ( yn+untv un , unt un ) tends to (y, t). Therefore (y, t)

), then there is a sequence of points (y n , t n ) n∈N in Y v and a sequence of positive real numbers (u n ) n∈N such that u n → 0 and ( yn un , tn un ) → (y, t). Then y n -t n v ∈ Y and yn-tnv un tends to y -tv. So y -tv belongs to

Let us assume that X is equipped with a Whitney stratification S = {S i } l i=0 with S 0 = {0} and 0 ∈ S i for i = 1, . . . , l. We set

for i = 0, . . . , s.

Lemma 8.8. The partition

Proof. The proof is the same as in the previous lemma, taking into account the following remark: if

For 0 < u ≤ 0 , we set X u = X ∩ S n-1 u and we denote by CX u the cone over X u , i.e.

CX u = {x ∈ R n | ∃λ ∈ R + and z ∈ X u such that x = λz}.

Lemma 8.12. Let us assume that C 0 X and C 0 Y satisfy Condition [START_REF] Bekka | Regular stratification of subanalytic sets[END_REF]. There exists a definable subset ∆ X,Y ⊂ S n-1 of positive codimension such that for v / ∈ ∆ X,Y ,

Proof. As in Lemma 8.1, we have to prove that

There exist v > 0 and a > 0 such that

which implies that there exists u 0 > 0 such that

We have assumed that C 0 X and C 0 Y satisfy Condition [START_REF] Bekka | Regular stratification of subanalytic sets[END_REF]. This means that two strata W and W of C 0 X and C 0 Y (different from {0}) intersect transversally. Since these strata are conic, W ∩ {ω = 1} and W intersect transversally as well and so C 0 X ∩ {ω = 1} and C 0 Y intersect transversally (in the stratified sense). As in Lemma 4.11, there exists a definable subset Γ C0X∩{ω=1},C0Y ⊂ S n-1 of positive codimension such that for v / ∈ Γ C0X∩{ω=1},C0Y , C 0 X ∩ {ω = 1} and (C 0 Y ) v intersect transversally (in the stratified sense).

We need a first auxiliary lemma.

Lemma 8.13. If v / ∈ Γ C0X∩{ω=1},C0Y , then there exists 0 < u 1 ≤ u 0 such that for 0 < u ≤ u 1 and for (x, t)

Proof. Let us specify the stratifications we are working with. The set Y is equipped with a Whitney stratification {T j } m j=0 , which induces a stratification

The set X is equipped with a Whitney stratification {S i } l i=0 . Hence for u small, CX u is stratified by

As above this induces a stratification of CX u ∩ {t > 0}. We note that by the conic structure, the intersection CX u ∩ {t > 0} ∩ {ω = ω(x)} is always transverse (in the stratified sense) and the stratification of

The same computation applied to X ∩ {u * = 0} yields

Λi (Lk(X ∩ {u * = 0}), Lk(X ∩ {u * = 0}))

Using the notations and normalizations of [START_REF] Bernig | Courbures intrinsèques dans les catégories analyticogéométriques[END_REF], Theorem 4.4, we can write

where E is a (n -2)-dimensional unit sphere in S n-1 . By the spherical kinematic formula, we find that

Hence we get that

Then we apply this result to CX u and CY u where

, and make u → 0 and obtain

Therefore by the relation between σ(X, Y, 0) and Λ 0 (X, Y, 0) and by Theorem 8.15, we get

where

By [START_REF] Dutertre | Stratified critical points on the real Milnor fibre and integral-geometric formulas[END_REF], Theorem 5.6 and its proof, we have that σ n (Y, 0) = Λ lim n (Y, 0) and for i ≥ 1,

Moreover by [START_REF] Dutertre | Euler characteristic and Lipschitz-Killing curvatures of closed semialgebraic sets[END_REF], Theorem 5.1, we have that

These equalities enable us to end the proof.

For Y = H, where H ∈ G n-k n and k ∈ {1, . . . , n}, the above kinematic formula writes σ(X, H, 0) = Λ lim k (X, 0). Hence we recover our Theorem 3.7, because for H ∈ G n-k n , β 0 (H) = 1 s n-1 S n-1 lim →0 lim δ→0 + Λ 0 (H δ,v ∩ X, H δ,v ∩ X ∩ B n )dv, by the co-area formula.

More kinematic formulas

In view of Theorem 8.17, a natural question is to express the following sums i+j=p+n Λ i (X, 0) • Λ j (X, 0), for k = 1, . . . , n as the right-hand side of a kinematic formula. The answer is quite simple and explained briefly in the next proposition. Proposition 9.1. Let (X, 0) ⊂ (R n , 0) and (Y, 0) ⊂ (R n , 0) be two germs of closed definable sets. For k ∈ {1, . . . , n}, the following kinematic formula holds: