Principal Kinematic Formulas for Germs of Closed Definable Sets
aa r X i v : . [ m a t h . AG ] D ec PRINCIPAL KINEMATIC FORMULAS FOR GERMS OF CLOSEDDEFINABLE SETS
NICOLAS DUTERTRE
Abstract.
We prove two principal kinematic formulas for germs of closeddefinable sets in R n , that generalize the Cauchy-Crofton formula for the densitydue 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 − . 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 [25]);- For manifolds by Chern [7] and manifolds with boundary by Santal´o [35];- For PL-sets by Cheeger, M¨uller and Schrader [6];- For sets with positive reach by Federer [19, 20] (see also [34]);- For subanalytic sets by Fu [22], and more generally for sets definable in ano-minimal structure by Bernig, Br¨ocker and Kuppe [4, 2, 3].There are many other situations where kinematic formulas hold, but we cannot givehere 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-minimalstructure. Definable sets are a generalization of semi-algebraic sets and globallysubanalytic sets, we refer the reader to classical references [37, 36, 10, 29, 33] forbasic definitions and results on this topics. The study of the geometric properties ofthese objects was initiated by Fu [22], who developed integral geometry for compactsubanalytic sets. Using the technology of the normal cycle, he associated with everycompact subanalytic set X of R n a sequence of curvature measuresΛ ( X, − ) , . . . , Λ n ( X, − ) , called the Lipschitz-Killing measures, and he established several integral geom-etry formulas. Among them, he proved the following kinematic formulas: for k ∈ { , . . . , n } , we have Z SO ( n ) ⋉R n Λ k ( X ∩ γY, U ∩ γV ) dγdx = X 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 areBorel subsets of X and Y respectively. We will state these formulas specifically Mathematics Subject Classification.
Key words and phrases.
Kinematic formulas, definable sets, Lipschitz-Killing curvatures, polarinvariants.The author is partially supported by the ANR project LISA 17-CE400023-01. in the next section. In [4] (see also [2, 3]), Br¨ocker and Kuppe gave a geometriccharacterization of these measures using stratified Morse theory, in the more generalsetting of definable sets.In [8] Comte started the study of real equisingularity by proving that the densityis continuous along the strata of a Verdier stratification of a subanalytic set (see also[38]). The main tool to prove his result was a local Cauchy-Crofton formula for thedensity. He continued this work with Merle in [9] where a similar continuity resultwas established for the so-called local Lipschitz-Killing invariants (see also [32]).The tools for proving this continuity property are local linear kinematic formulasthat generalize the Cauchy-Crofton formula for the density. These formulas willbe explained in Section 3 but, roughly speaking, they relate the so-called polarinvariants, which are mean-values of Euler characteristics of real Milnor fibres ofgeneric projections, to the local Lipschitz-Killing invariants.In [15] we also established an infinitesimal linear kinematic formula. It is slightlydifferent from the ones of Comte and Merle, because instead of using projections,we make “infinitesimally small” translations of linear spaces. Let us recall it herebecause 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 ofthe unit ball B k of dimension k , • for k ∈ { , . . . , n } , G kn is the Grassmann manifold of k -dimension linearspaces in R n equipped with the O ( n )-invariant Maurer-Cartan density (seefor instance [35], p.200), g kn is its volume, • if P is a linear subspace of R n of dimension k , S k − P is the unit sphere in P , • in R n , B nǫ ( x ) is the closed ball of radius ǫ centered at x and S n − ǫ ( x ) is thesphere of radius ǫ centered at x , if x = 0, we simply write B nǫ and S n − ǫ .Let ( X, ⊂ ( R n ,
0) be the germ of a closed definable set. We consider the followinglimits: Λ lim k ( X,
0) := lim ǫ → Λ k ( X, X ∩ B nǫ ) b k ǫ k . Let H ∈ G n − kn , k ∈ { , . . . , n } , and let v be an element in S k − H ⊥ . For δ >
0, wedenote by H v,δ the ( n − k )-dimensional affine space H + δv and we set β ( H, v ) = lim ǫ → lim δ → Λ ( H δ,v ∩ X, H δ,v ∩ X ∩ B nǫ ) , and β ( H ) = 1 s k − Z S k − H ⊥ β ( H, v ) dv. In [15] Theorem 5.5, we proved that for k ∈ { , . . . , n } Λ lim k ( X,
0) = 1 g n − kn Z G n − kn β ( 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 definableset? 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, ⊂ ( R n ,
0) be the germ ofa closed definable set. To such a germ, we associate two sequences of real numbers:
RINCIPAL KINEMATIC FORMULAS FOR GERMS OF CLOSED DEFINABLE SETS 3 the polar invariants σ k ( X, k = 0 , . . . , n , and the above limits Λ lim k ( X, Y, ⊂ ( R n ,
0) be another germ of closed definable set and let σ ( X, Y,
0) = 1 s n − Z SO ( n ) × S n − lim ǫ → lim δ → + χ ( X ∩ ( γY + δv ) ∩ B nǫ ) dγdv. Here SO ( n ) is equipped with the Haar measure dγ , normalized in such a waythat the volume of SO ( n ) is s n − , S n − is equipped with the usual Riemanniannmeasure (or density) dv and SO ( n ) × S n − with the product measure dγdv . Ourfirst infinitesimal principal kinematic formula takes the following form (Theorem8.15): σ ( X, Y,
0) = n X i =0 Λ lim i ( X, · σ n − i ( Y, . 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 andthe right-hand side is the product of the densities of X and Y at the origin (seeCorollary 8.16). Then we setΛ lim0 ( X, Y, s n − Z SO ( n ) × S n − lim ǫ → lim δ → + Λ ( X ∩ ( γY + δv ) , X ∩ ( γY + δv ) ∩ B nǫ ) dγdv. In Theorem 8.17, we establish our second infinitesimal principal kinematic for-mula: Λ lim0 ( X, Y,
0) = n X i =0 Λ lim i ( X, · Λ lim n − i ( Y, . This formula is a corollary of Theorem 8.15 and the Gauss-Bonnet formula provedin [15].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 ) = h v, y i , • 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 ) itsEuler characteristic for Borel-Moore homology.The paper is organized as follows. In Section 2, we recall the notion of strat-ified critical points and the definition of the Lipschitz-Killing measures. We alsostate kinematic formulas. In Section 3, we recall the Gauss-Bonnet formula for realMilnor fibres proved by the author in [15], and the infinitesimal linear kinematicformulas proved by Comte [8], Comte and Merle [9] and the author [15]. Section 4contains several topological and geometrical lemmas that will be useful in the nextsections. In Section 5, we prove a new spherical kinematic formula for definablesets. Combining this formula with Hardt’s theorem [24, 10, 37], we obtain a newkinematic formula for definable subsets of the unit ball in Section 6. We applythis formula in Section 7 to get our first principal kinematic formula for closedconic definable sets. In Section 8, we prove our first principal kinematic formulain the general case using the previous case and tangent cones, and then our sec-ond principal kinematic formula. Finally Section 9 contains two other kinematicformulas. NICOLAS DUTERTRE Stratified critical points and Lipschitz-Killing curvatures
Stratified critical points.
Let X ⊂ R n be a compact definable set equippedwith a finite definable Whitney stratification S = { S a } a ∈ A . The fact that such astratification exists is due to Loi [28] (see also [31]).Let f : X → R be a definable function. We assume that f is the restriction to X of a C 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 indexof f at p is defined byind( f, X, p ) = 1 − χ ( X ∩ { f = f ( p i ) − δ } ∩ B nε ( 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’stheorem [24, 10, 37]. Theorem 2.2.
Assume that f : X → R has a finite number of critical points { p , . . . , p s } . Then the following equality holds: χ ( X ) = s X i =1 ind( f, X, p i ) . Proof.
See Theorem 3.1 in [13]. When f is a Morse stratified function, this followsfrom [23]. (cid:3) Lipchitz-Killing curvatures.
In this subsection, we present the Lipschitz-Killing measures of a definable set in an o-minimal structure. We describe Br¨ockerand Kuppe’s approach [4].Let X ⊂ R n be a compact definable set equipped with a finite definable Whitneystratification S = { S a } a ∈ A .Let us fix a stratum S . For k ∈ { , . . . , d S } , d S = dim S , let λ Sk : S → R bedefined by λ Sk ( x ) = 1 s n − k − Z S TxS ⊥ 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 (cid:16) X ∩ N x ∩ B nǫ ( x ) ∩ { v ∗ = v ∗ ( x ) − δ } (cid:17) , where 0 < δ ≪ ǫ ≪ 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 Morseindex at x of a function f : R n → R such that f | X has a stratified Morse criticalpoint at x and ∇ f ( x ) = v . For k ∈ { d S + 1 , . . . , n } , we set λ Sk ( x ) = 0.If S has dimension n then for all x ∈ S , we put λ S ( x ) = · · · = λ Sn − ( x ) = 0 and λ Sn ( x ) = 1. If S has dimension 0 then ind nor ( v ∗ , X, x ) = ind( v ∗ , X, x ) and we set λ S ( x ) = 1 s n − Z S n − ind( v ∗ , X, x ) dv, and λ Sk ( x ) = 0 if k > RINCIPAL KINEMATIC FORMULAS FOR GERMS OF CLOSED DEFINABLE SETS 5
Definition 2.3.
For every Borel set U ⊂ X and for every k ∈ { , . . . , n } , we defineΛ k ( X, U ) by Λ k ( X, U ) = X a ∈ A Z S a ∩ U λ S a k ( x ) dx. These measures Λ k ( X, − ) are called the Lipschitz-Killing measures of X . Notethat for any Borel set U of X , we haveΛ 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 ∈ { , . . . , d } , Λ k ( X, U )is equal to 1 s n − k − Z 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 ( X, − ) satisfies an exchange formula (see[4]). Proposition 2.4.
For every Borel set U ⊂ X , we have Λ ( X, U ) = 1 s n − Z S n − X x ∈ U ind( v ∗ , X, x ) dv. For U = X and by Theorem 2.2, we see that a special case of this exchangeformula is the Gauss-Bonnet formula Λ ( X, X ) = χ ( X ).The Lipschitz-Killing measures satisfy the kinematic formula (see [22, 4, 2, 3]).We provide the group SO ( n ) ⋉ R n of all euclideans motions of R n with the prod-uct measure dγdx , where the canonical Haar measure dγ is normalized such thatvol ( 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 ∈ { , . . . , n } , the following kinematicformula holds: Z SO ( n ) ⋉R n Λ k ( X ∩ γY, U ∩ γV ) dγdx = X p + q = k + n e ( p, q, n )Λ p ( X, U )Λ q ( X, V ) , where e ( p, q, n ) = s p + q − n s n s p s q . For k = 0, the above formula is called the principal kinematic formula. Aparticular case of the kinematic formula is the linear kinematic formula. Let A kn bethe affine grassmannian of k -dimensional affine spaces in R n . It is a fibre bundleover G kn with fibre R n − k . We equip A kn with the product measure denoted by dE . Proposition 2.6.
Let X ⊂ R n be a compact definabet set and let U ⊂ X be aBorel set. For k ∈ { , . . . , n } , we have Λ n − k ( X, U ) = 1 g kn · e ( k, n − k, n ) Z A kn Λ ( X ∩ E, X ∩ E ∩ U ) dE. In Section 5, we will consider definable subsets of the unit sphere S n − . For suchsets, one can define spherical Lipschitz-Killing measures. These measure are definedin [3], Section 3 (see also [14]). Their definition is very similar to the definition ofthe above Lipschitz-Killing measures. For X ⊂ S n − and k ∈ { , . . . , n − } , we will NICOLAS DUTERTRE denote by ˜Λ k ( X, − ) the k -th spherical Lispchitz-Killing measures. The sphericalLipschitz-Killing measures satisfy a Gauss-Bonnet formula ([3], Theorem 1.2) anda spherical kinematic formula ([3], Theorem 4.4).3. Some topological ang geometrical properties of definable sets
In this section, we review some results on the local topology and geometry ofclosed definable sets. Let ( X,
0) be the germ of a closed definable set. For conve-nience, we will work with a small representative that we denote by X as well. Weassume 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 possibletaking a smaller representative if necessary).Let ρ i : U → R , i = 1 ,
2, be two continuous definable functions of class C on U \ { } , such that ρ − i (0) = { } and ρ i ( x ) ≥ x ∈ X . It is well-known that there exists ǫ i > < ǫ ≤ ǫ i , ρ − i ( ǫ ) intersects X transversally in the stratified sense (see [15] Lemma 2.1), and that the topologicaltype of ρ − i ( ǫ ) ∩ X does not depend on ǫ . Moreover, as explained by Durfee in [11],Lemma 1.8 and Corollary 3.6, there is a neighborhood Ω of 0 in R n such that forevery stratum S of X , ∇ ( ρ | S ) and ∇ ( ρ | S ) do not point in opposite direction inΩ \ { } . Applying Durfee’s argument ([11], Proposition 1.7 and Proposition 3.5),we see that ρ − ( ǫ ) ∩ X , 0 < ǫ ≤ ǫ , and ρ − ( ǫ ′ ) ∩ X , 0 < ǫ ′ ≤ ǫ , are homeomorphic.The link of X at 0, denoted by Lk( X ), is the set X ∩ ρ − ( ǫ ), 0 < ǫ ≪
1, where ρ : U → R is a continuous definable function of class C on U \ { } , such that ρ − (0) = { } and ρ ( x ) ≥ x ∈ X . We will call such a function ρ a distancefunction to the origin. By the above discussion, the topological type of Lk( X ) doesnot depend on the choice of the definable distance function to the origin (actuallyto define the link, we do not need to assume that ρ is C on U \ { } , continuity isenough).Let f : ( X, → ( R ,
0) be the germ of a definable function. We assume that f is the restriction to X of a C definable function F : U → R . We denote by X f the set f − (0) and by [1, 27], 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 Whitneystratification 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, → ( 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 thefollowing stratification: (cid:8) S α \ f − (0) , S α ∩ ( f − (0) \ { } ) , { } | α ∈ A (cid:9) , is a Thom stratification of X adapted to X f .As explained above, there is ǫ ′ > < ǫ ≤ ǫ ′ , ρ − ( ǫ ) intersects X f transversally. The Thom condition implies that there exists δ ǫ > δ with 0 < δ ≤ δ ǫ , ρ − ( ǫ ) intersects f − ( δ ) transversally as well. Hencethe set f − ( δ ) ∩ { ρ ≤ ǫ } is a Whitney stratified set equipped with the followingstratification: (cid:8) f − ( δ ) ∩ V β ∩ { ρ < ǫ } , f − ( δ ) ∩ V β ∩ { ρ = ǫ } | V β * X f (cid:9) . RINCIPAL KINEMATIC FORMULAS FOR GERMS OF CLOSED DEFINABLE SETS 7
Moreover, taking ǫ ′ and δ ǫ smaller if necessary, the topological types of f − ( δ ) ∩{ ρ ≤ ǫ } and f − ( δ ) ∩ { ρ = ǫ } do not depend on the couple ( ǫ, δ ). To see this, itis enough to adapt the proof of Lemma 2.1 in [18] to the stratified case. The samefact is true for negative values of f .Of course, we can make the same construction with ρ instead of ρ . But asabove, there is a neighborhood Ω ′ of 0 in R n such that for every stratum W of X f , ∇ ( ρ | W ) and ∇ ( ρ | W ) do not point in opposite direction. Let us choose ǫ ′ > ǫ > { ρ ≤ ǫ ′ } ( { ρ ≤ ǫ } ⊂ Ω ′ . If ǫ , ǫ ′ and δ are sufficientlysmall then, for every stratum V * X f , ∇ ( ρ | V ∩ f − ( δ ) ) and ∇ ( ρ | V ∩ f − ( δ ) ) do notpoint in opposite direction in { ρ ≤ ǫ } \ { ρ < ǫ ′ } . Otherwise, by Thom ( a f )-condition, we would find a point p in X f ∩ ( { ρ ≤ ǫ } \ { ρ < ǫ ′ } ) such thateither ∇ ( ρ | W )( p ) or ∇ ( ρ | W )( p ) vanish or ∇ ( ρ | W )( p ) and ∇ ( ρ | W )( p ) point inopposite direction, where W is the stratum of X f that contains p (see the proofof Lemma 3.7 in [15]). This is impossible if we are sufficiently close to the origin.Applying Durfee’s argument mentioned above, we see that f − ( δ ) ∩ { ρ ≤ ǫ } ishomeomorphic to f − ( δ ) ∩ { ρ ≤ ǫ ′ } and that f − ( δ ) ∩ { ρ = ǫ } is homeomorphicto f − ( δ ) ∩ { ρ = ǫ ′ } .The positive (resp. negative) Milnor fibre of f is the set f − ( δ ) ∩ { ρ ≤ ǫ } (resp. f − ( − δ ) ∩ { ρ ≤ ǫ } ), where 0 < δ ≪ ǫ ≪ ρ is a distance function to the origin.The set f − ( ± δ ) ∩ { ρ = ǫ } is the boundary of the Milnor fibre. By the previousdiscussion, the topological type of the positive (resp. negative) Milnor fibre andthe topological type of its boundary do not depend on the choice of the definabledistance function to the origin.In [15], we considered a second definable function-germ g : ( R n , → ( R ,
0) andwe assumed that g was the restriction to X of a C definable function G : U → R .Moreover, we assumed that g satisfied the following two conditions: • Condition (A): g : ( X, → ( R ,
0) has an isolated critical point at 0. • Condition (B): the relative polar setΓ f,g = ⊔ V β * X f Γ V β f,g = ⊔ V β * X f (cid:8) x ∈ V β | rank (cid:2) ∇ ( f | V β )( x ) , ∇ ( g | V β )( x ) (cid:3) < (cid:9) is a 1-dimensional C definable set (possibly empty) in a neighborhood ofthe origin.We wrote Γ f,g = ⊔ li =1 B i , where each B i is a definable connected curve, and weconsidered the intersections points of Γ f,g with f − ( δ ) ∩ B nǫ :Γ f,g ∩ ( f − ( δ ) ∩ B nǫ ) = ⊔ li =1 B i ∩ ( f − ( δ ) ∩ B nǫ ) = n p δ,ǫ , . . . , p δ,ǫr o , where 0 < | δ | ≪ ǫ ≪
1. The points p δ,ǫi are exactly the critical points of g on f − ( δ ) ∩ ˚ B nǫ . Then we set I ( δ, ǫ, g ) = r X i =1 ind( g, f − ( δ ) , p δ,ǫi ) ,I ( δ, ǫ, − g ) = r X i =1 ind( − g, f − ( δ ) , p δ,ǫi ) , and in [15], Theorem 3.10, we related I ( δ, ǫ, g ) + I ( δ, ǫ, − g ), with 0 < | δ | ≪ ǫ ≪ NICOLAS DUTERTRE
Let us give now a new characterization of I ( δ, ǫ, g ) and I ( δ, ǫ, − g ) independenton δ and ǫ . Let us fix a connected component B of Γ f,g . We can assume that f isstricly increasing on B and we put B ∩ f − ( δ ) = { p δ } for δ > Lemma 3.1.
There exists δ > such that for < δ ≤ δ , the function δ ind( g, f − ( δ ) , p δ ) is constant on ]0 , δ ] .Proof. Let d : R n → R be the distance function to B . It is a continuous definablefunction on an open definable neighborhood O of B . Let 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 be the mapping definedby ρ ( x ) = ( f ( x ) , d ( x )). By Hardt’s theorem [24, 10, 37], there is a partition of]0 , + ∞ [ × ]0 , + ∞ [ into finitely many definable sets such that ρ is trivial over each ofthis set. Let us denote by ∆ the union of the members of this partition which havedimension 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 { ǫ ′ | ( ν, ǫ ′ ) ∈ ∆ } is definable and bythe previous remark, there is ν > r ( ν ) > < ν < ν . Hence bythe Monotonicity Theorem (see [10], Theorem 2.1 or [37], 4.1), there is 0 < δ < ν such that r is continuous, monotone and strictly positive on ]0 , δ ]. Moreover thefunction ( δ, ǫ ′ ) χ ( A ∩ { f = δ } ∩ { d = ǫ ′ } ) is constant on { ( δ, ǫ ′ ) | < δ < δ , <ǫ ′ < r ( δ ) } . But, by Lemma 3.1 in [18] and the above discussion on the topology ofthe link, we haveind( g, f − ( δ ) , p δ ) = 1 − χ (cid:0) { g ≤ g ( p δ ) } ∩ { f = δ } ∩ { d = ǫ ′ } (cid:1) = 1 − χ ( A ∩ { f = δ } ∩ { d = ǫ ′ } ) . We conclude that the function δ ind( g, f − ( δ ) , p δ ) is constant on ]0 , δ ]. (cid:3) Of course, a similar result holds for negative values of f .By the general Lojasiewicz inequality (see [4], Corollary 1.5.2), there exists acontinuous definable function ψ : ( R , → ( R ,
0) such that | p | ≤ ψ ( f ( p )) for p ∈ B .Moreover ψ is of class C in an open neighborhood of 0 and ψ ( u ) > u > ǫ > < δ < ψ − ( ǫ ) then | p | ≤ ǫ for p ∈ B ∩ f − ( δ ).Since Γ f,g consists of a finite number of branches, we can conclude that for ǫ > δ > < | δ | ≤ δ , Γ f,g ∩ f − ( δ ) ⊂ B n ǫ , andso Γ f,g ∩ ( f − ( δ ) ∩ B nǫ ) = Γ f,g ∩ f − ( δ ). With the above notation, this meansthat p δ,ǫi = p δi for 0 < | δ | ≪ ǫ ≪ i ∈ { , . . . , r } . For i ∈ { , . . . , r } , let τ i ( g ) (resp. τ i ( − g )) be the value that the function δ ind( g, f − ( δ ) , p δi ) (resp.ind( − g, f − ( δ ) , p δi )) takes close to the origin. We deduce the following relations:lim ǫ → lim δ → + I ( δ, ǫ, g ) + I ( δ, ǫ, − g ) = X i | f> on B i τ i ( g ) + τ i ( − g ) , lim ǫ → lim δ → − I ( δ, ǫ, g ) + I ( δ, ǫ, − g ) = X i | f< on B i τ i ( g ) + τ i ( − g ) . Of course, the same study can be done with another definable distance func-tion to the origin and so, the two limits lim ǫ → lim δ → + I ( δ, ǫ, g ) + I ( δ, ǫ, − g ) andlim ǫ → lim δ → − I ( δ, ǫ, g ) + I ( δ, ǫ, − g ) do not depend on the distance function to theorigin chosen to define the Milnor fibre of f . RINCIPAL KINEMATIC FORMULAS FOR GERMS OF CLOSED DEFINABLE SETS 9
Applying this study to a linear form v ∗ , where v is generic in S n − , we establishedin [15], 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 criticalpoint at 0. Namely if X is equipped with a Whitney stratification for which f :( X, → ( R ,
0) has an isolated stratified critical point at 0, then ([15], Corollary4.6) lim ǫ → lim δ →± Λ (cid:0) f − ( δ ) , f − ( δ ) ∩ B nǫ (cid:1) = lim ǫ → lim δ →± χ (cid:0) f − ( δ ) ∩ B nǫ (cid:1) − χ (cid:0) Lk( X f ) (cid:1) − s n − Z S n − χ (cid:0) Lk( X f ∩ { v ∗ = 0 } ) (cid:1) dv. The proof of this Gauss-Bonnet formula relies on the following exchange formula:lim ǫ → lim δ →± Λ (cid:0) f − ( δ ) , f − ( δ ) ∩ B nǫ (cid:1) = 12 s n − Z S n − lim ǫ → lim δ →± (cid:2) I ( δ, ǫ, v ∗ ) + I ( δ, ǫ, − v ∗ ) (cid:3) dv. But we have explained above that lim ǫ → lim δ → I ( δ, ǫ, v ∗ ) + I ( δ, ǫ, − v ∗ ) does notdepend on the choice of the distance function to the origin used to define the Milnorfibre of f . Therefore the relations proved in [15], Theorem 4.5 and Corollary 4.6,are also valid if we replace the usual euclidian distance function by any definabledistance 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 usrecall the definition of the polar invariants [9]. Let k ∈ { , . . . , n } and let P ∈ G kn .We denote by π P : X → P the orthogonal projection on P . For P generic in G kn , Comte and Merle established the existence of an open and dense definablegerm ( K P , ⊂ ( P,
0) such that, if K P = ∪ N P i =1 K Pi denotes its decomposition intoconnected components, then the function K Pi χ Pi := lim ǫ → lim y ∈ KPiy → χ (cid:0) π − P ( y ) ∩ X ∩ B nǫ (cid:1) is well-defined. Then they set the following definition: Definition 3.2.
Let k ∈ { , . . . , n } . The polar invariant σ k ( X,
0) is defined by σ k ( X,
0) = 1 g kn Z G kn N P X i =1 χ Pi · Θ( K pi , dP. We set σ ( X,
0) = 1.In [9] the authors defined another sequence of invariants attached to X , calledthe local Lipschitz-Killing invariants. Definition 3.3.
Let k ∈ { , . . . , n } . The local Lipschitz-Killing invariant Λ loc k ( X, loc k ( X,
0) = lim ǫ → Λ k ( X ∩ B nǫ , X ∩ B nǫ ) b k ǫ k . We note that Λ loc0 ( X,
0) = 1. Then Comte and Merle proved linear kinematicformulas that relate the local Lipschitz-Killing invariants to the polar invariants.
Theorem 3.4 ([9], Theorem 3.1) . For any germ ( X, ⊂ ( R n , of definable closedset, we have Λ loc1 ( X, ... Λ loc n ( X, = m . . . m n . . . m n ... ... . . . ... . . . · σ ( X, ... σ n ( X, , where m ji = b j b j − i b i (cid:0) ji (cid:1) − b j − b j − − i b i (cid:0) j − i (cid:1) , for i + 1 ≤ j ≤ n . If dim X = d then σ d +1 ( X,
0) = · · · = σ n ( X,
0) = 0 and one recovers the localCauchy-Crofton formula Λ loc d ( X,
0) = σ d ( X, k = 0 , . . . , n , we considered the limitsΛ lim k ( X,
0) := lim ǫ → Λ k ( X, X ∩ B nǫ ) b k ǫ k , and we showed the following theorem: Theorem 3.5 ([14], Theorem 5.1) . For any germ ( X, ⊂ ( R n , of definableclosed set, we have Λ lim0 ( X,
0) = 1 − χ (Lk( X )) − g n − n Z G n − n χ (Lk( X ∩ H )) dH. Furthermore for k ∈ { , . . . , n − } , we have Λ lim k ( X,
0) = − g n − k − n Z G n − k − n χ (Lk( X ∩ H )) dH + 12 g n − k +1 n Z G n − k +1 n χ (Lk( X ∩ L )) dL, and: Λ lim n − ( X,
0) = 12 g n Z G n χ (Lk( X ∩ H )) dH, Λ lim n ( X,
0) = 12 g n Z G n χ (Lk( X ∩ H )) dH. As a corollary, we obtained:
Corollary 3.6 ([14], Corollary 5.2) . For any germ ( X, ⊂ ( R n , of definableclosed set, the equality P nk =0 Λ lim k ( X, holds. We note that Λ lim k ( X,
0) differs from Λ loc k ( X, X does nothave any contribution in the computation of Λ lim k ( X, lim k ( X, H ∈ G n − kn , k ∈ { , . . . , n } , and let v be an element in S k − H ⊥ . For δ >
0, we denote by H v,δ the ( n − k )-dimensional affine space H + δv and we set β ( H, v ) = lim ǫ → lim δ → Λ ( H δ,v ∩ X, H δ,v ∩ X ∩ B nǫ ) . Then we set β ( H ) = 1 s k − Z S k − H ⊥ β ( H, v ) dv. RINCIPAL KINEMATIC FORMULAS FOR GERMS OF CLOSED DEFINABLE SETS 11
Theorem 3.7 ([15], Theorem 5.5) . For k ∈ { , . . . , n } , we have Λ lim k ( X,
0) = 1 g n − kn Z G n − kn β ( H ) dH. We also proved local linear kinematic formulas that relate the limits Λ lim k ( X, Theorem 3.8 ([15], Theorem 5.6) . For k ∈ { , . . . , n − } , we have Λ lim k ( X,
0) = σ k ( X, − σ k +1 ( X, . Furthermore, we have Λ lim n ( X,
0) = σ n ( X, . 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 openneighborhood U of 0.4.1. A Gauss-Bonnet formula.
Let { S i } li =0 be a Whitney stratification of X ,where S = { } and 0 ∈ S i for i ∈ { , . . . , l } . Similarly let { T j } mj =0 be a Whitneystratification of Y , where T = { } and 0 ∈ T j for j = { , . . . , m } . We assume eachstratum to be connected. We introduce the following condition: • Condition (1): for i ∈ { , . . . , l } and for j ∈ { , . . . , m } , S i and T j intersecttransversally (the case S i ∩ T j = ∅ is possible).If X and Y satisfy Condition (1) then X ∩ Y admits a Whitney stratification X ∩ Y = ⊔ rk =0 R k , where R = { } and each R k is a connected component of anintersection S i ∩ T j , ( i, j ) ∈ { , . . . , l } × { , . . . , m } .Let b X ⊂ R n +1 be the following definable set: b X = (cid:8) ( x, t ) ∈ R n +1 | x ∈ X (cid:9) . It is included and closed in U × R . Let v ∈ S n − and let c Y v ⊂ R n +1 be the followingdefinable set: c Y v = (cid:8) ( y, t ) ∈ R n +1 | ∃ y ′ ∈ Y such that y = y ′ + tv (cid:9) . It is included and closed in the open set c U v = (cid:8) ( u, t ) ∈ R n +1 | ∃ u ′ ∈ U such that u = u ′ + tv (cid:9) . It is well-known (see [23]) that we can equip b X with a Whitney stratification in thefollowing way: b X = ⊔ li =0 b S i where b S i = { ( x, t ) ∈ R n +1 | x ∈ S i } . Similarly we canconsider the following partition of c Y v : c Y v = ⊔ mj =0 [ ( T j ) v where [ ( T j ) v = (cid:8) ( y, t ) ∈ R n +1 | ∃ y ′ ∈ T j such that y = y ′ + tv (cid:9) . Lemma 4.1.
The partition c Y v = ⊔ mj =0 [ ( T j ) v gives a Whitney stratification of c Y v .Proof. With obvious notations, the partition b Y = ⊔ mj =0 c T j induces a Whitney strat-ification of b Y . Let φ : U × R → c U v be defined by φ ( u, t ) = ( u + tv, t ). Then φ isa diffeomorphism, φ ( b Y ) = c Y v and φ ( c T j ) = [ ( T j ) v for j ∈ { , . . . , m } . This gives theresult for Whitney’s conditions are invariant by C -diffeomorphisms. (cid:3) From now on, we will focus on the definable set b X ∩ c Y v . Let us denote if by Z v .It is included and closed in the open set ( U × R ) ∩ c U v . We introduce the followingsecond condition: • Condition (2): for i ∈ { , . . . , l } and j ∈ { , . . . , m } , the strata b S i and [ ( T j ) v intersect transversally outside (0 , v satisfies Condition (2) then Z v admits a Whitney stratification Z v = ⊔ ql =0 Q l where each Q l is a connected component of an intersection b S i ∩ [ ( T j ) v . We note thatnecessarly c S ∩ [ ( T ) v = { (0 , } and that we can put Q = { (0 , } . Lemma 4.2.
Assume that X and Y satisfy Condition (1) and that v satisfiesCondition (2). Then the function t | Z v : Z v → R ( y, t ) t has an isolated stratified critical point at (0 , .Proof. Let Q be a stratum of Z v different from { (0 , } . Since the critical points of t | Q lie in { t = 0 } , we can suppose that Q is a connected component of b S i ∩ [ ( T j ) v with i = 0 and j = 0. Let us prove that { t = 0 } intersects b S i ∩ [ ( T j ) v transversally.If it is not the case, then there is a point p in b S i ∩ [ ( T j ) v ∩ { t = 0 } such that T p ( b S i ∩ [ ( T j ) v ) ⊂ R n . But it is not difficult to check that { t = 0 } intersects b S i and [ ( T j ) v transversally, so T p S i = T p b 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 ( b S i ∩ [ ( T j ) v ) = T p b S i ∩ T p [ ( T j ) v . We get that T p ( S i ∩ T j ) = T p ( b S i ∩ [ ( T j ) v ). Thisis not possible, for dim b S i = dim S i + 1, dim [ ( T j ) v = dim T j + 1 and these twointersections are transverse in R n and R n +1 . (cid:3) We can apply Corollary 4.6 in [15] to t and Z v . Corollary 4.3.
Assume that X and Y satisfy Condition (1) and that v satisfiesCondition (2). Then we have lim ǫ → lim δ →± Λ (cid:0) Z v ∩ { t = δ } , Z v ∩ { t = δ } ∩ B n +1 ǫ (cid:1) = lim ǫ → lim δ →± χ (cid:0) Z v ∩ { t = δ } ∩ B n +1 ǫ (cid:1) − χ (Lk( X ∩ Y )) − s n − Z S n − χ (Lk( X ∩ Y ∩ { u ∗ = 0 } ) du. Proof.
We just have to show that12 s n − Z S n − χ (Lk( X ∩ Y ∩ { u ∗ = 0 } ) du = 12 s n Z 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[12] applies here. (cid:3) 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 ǫ > such that for < ǫ ≤ ǫ , there exists δ ǫ > suchthat for < δ ≤ δ ǫ , the topological type of Z v,δ ∩ B nǫ does not depend on the choice RINCIPAL KINEMATIC FORMULAS FOR GERMS OF CLOSED DEFINABLE SETS 13 of the couple ( ǫ, δ ) . Moreover for < δ ≪ ǫ ≪ , Z v,δ ∩ B nǫ and Z v ∩ { t = δ } ∩ B n +1 ǫ are homeomorphic.Proof. Let ρ ( x, t ) = p x + · · · + x n + t . Then Z v,δ ∩ B nǫ is homeomorphic to Z v ∩{ t = δ } ∩ { ρ ≤ ǫ + δ } . Let π : Z v ∩ { t ≥ } → R be the mapping defined by π ( x, t ) = ( ρ ( x, t ) , t ) and let ∆ ⊂ R be its (stratified) discriminant. It is a definablecurve 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 ) > ν > r is continuous and increasing on a smallinterval ]0 , ν [. Let ( ǫ, δ ) be a couple such that 0 < ǫ < ν , 0 < δ < r ( ǫ ) and Z v ∩ { t = δ } ∩ { ρ ≤ ǫ } has the topological type of the Milnor fibre of t | Z v . Bytaking δ smaller if necessary, we can assume that ǫ + δ < ν . Since r is increasing, δ < r ( ǫ ) < r ( ǫ + δ ) and so Z v ∩ { t = δ } ∩ { ρ ≤ ǫ + δ } is homeomorphic to Z v ∩ { t = δ } ∩ { ρ ≤ ǫ } . We conclude with the results of Section 3. (cid:3) A similar result is true for negative values of t replacing p x + · · · + x n + t with p x + · · · + x n − t . We can state the infinitesimal Gauss-Bonnet formula for Z v,δ . Lemma 4.5.
Assume that X and Y satisfy Condition (1) and that v satisfiesCondition (2). Then we have lim ǫ → lim δ →± Λ ( Z v,δ , Z v,δ ∩ B nǫ ) = lim ǫ → lim δ →± χ ( Z v,δ ∩ B nǫ ) − χ (Lk( X ∩ Y )) − s n − Z S n − 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 Theorem5.0 in [22] or Proposition 9.2 in [4], we haveΛ n ( Z v,δ , Z v,δ ∩ B nǫ ) = Λ n +10 ( Z v ∩ { t = δ } , Z v ∩ { t = δ } ∩ { ρ ≤ ǫ + δ } ) . Here we suppose that δ >
0, Λ n (resp. Λ n +10 ) stands for the Gauss-Bonnet measurein R n (resp. R n +1 ) and ρ ( x, t ) = p x + · · · + x n + t As explained in Section 3, for u generic in S n and for ǫ > δ ǫ,u such that for 0 < δ ≤ δ ǫ,u , the critical points of u ∗ and − u ∗ in Z v,δ ∩ { t = δ } actually lie in Z v,δ ∩ { t = δ } ∩ { ρ ≤ ǫ } , hence there are not in Z v,δ ∩ { t = δ } ∩ { ǫ ≤ ρ ≤ ǫ + δ } . Thanks to this observation, we can conclude thatlim ǫ → lim δ → Λ n +10 ( Z v ∩ { t = δ } , Z v ∩ { t = δ } ∩ { ρ ≤ ǫ + δ } )= lim ǫ → lim δ → Λ n +10 ( Z v ∩ { t = δ } , Z v ∩ { t = δ } ∩ { ρ ≤ ǫ } ) . It is enough to apply Corollary 4.3 and the comments of Section 3 on the choice ofthe distance function to the origin to get the result. (cid:3)
A useful lemma.
We continue this section with a remark. Instead of trans-lating Y , we can translate X , intersect this translated set with Y and obtain anotherMilnor fibre Y ∩ ( X + δv ) ∩ B nǫ , 0 < | δ | ≪ ǫ ≪ Lemma 4.6.
There exists ǫ > such that for < ǫ ≤ ǫ , there exists δ ǫ > suchthat < δ ≤ δ ǫ , X ∩ ( Y + δv ) ∩ B nǫ and Y ∩ ( X − δv ) ∩ B nǫ are homeomorphic. Proof.
Let ρ v ( x, t ) = p ( x − tv ) + · · · + ( x n − tv n ) + t . By the results of Section3, we know that there exists ǫ > < ǫ ≤ ǫ , there exists δ ǫ > < δ ≤ δ ǫ , the topological type of Z v ∩ { t = δ } ∩ { ρ v ≤ ǫ } does notdepend on the couple ( ǫ, δ ) and is the topological type of the positive Milnor fibreof t | Z v . On the other hand, the set X ∩ ( Y + δv ) ∩ ( B nǫ + δv ) is homeomorphic to Z v ∩ { t = δ } ∩ { ρ v ≤ ǫ + δ } . The same method as the one used in Lemma 4.4 showsthat for 0 < ǫ ≤ ǫ and 0 < δ ≤ δ ǫ small enough, Z v ∩ { t = δ } ∩ { ρ v ≤ ǫ + δ } is homeomorphic to Z v ∩ { t = δ } ∩ { ρ v ≤ ǫ } . We conclude that Z v,δ ∩ B nǫ ishomeomorphic to Z v,δ ∩ ( B nǫ + δv ) for 0 < δ ≪ ǫ ≪
1. But h X ∩ ( Y + δv ) ∩ ( B nǫ + δv ) i − δv = ( X − δv ) ∩ Y ∩ B nǫ . (cid:3) Genericity of Conditions (1) and (2).
We prove the genericity of Condi-tions (1) and (2). To prove the genericity of Condition (1), we need some auxiliarylemmas.
Lemma 4.7.
Let x ∈ R n be a non-zero vector. We have (cid:8) Hx | H ∈ M n ( R ) such that t H = − H (cid:9) = x ⊥ . Proof.
It is clear that if H is an antisymmetric matrix, then Hx ∈ x ⊥ . Let us write x = ( x , . . . , x n ) and let a = ( a , . . . , a n ) ∈ x ⊥ . Since x = (0 , . . . ,
0) then thereexists 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 . (cid:3) Lemma 4.8.
Let f : R n → R k with ≤ k ≤ n − be a C mapping and let F bethe 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 issurjective 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 (cid:18) AY (cid:19) + t (cid:18) AY (cid:19) A = Y. Let β = Df ( Ax )( 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 . (cid:3) Lemma 4.9.
Let T ⊂ R n be a C definable submanifold of dimension d such that ∈ T . Then there exists a neighborhood U T of such that for x ∈ T ∩ ( U T \ { } ) , dim( T x T ∩ x ⊥ ) ≤ d − . RINCIPAL KINEMATIC FORMULAS FOR GERMS OF CLOSED DEFINABLE SETS 15
Proof.
If it is not the case, then there exists an injective C 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 h µ ( t ) , µ ′ ( t ) i = 0. This implies that h µ ( t ) , µ ( t ) i ′ = 0and that | µ ( t ) | = 0, which is not possible. (cid:3) Lemma 4.10.
There exists a definable subset Σ X,Y ⊂ SO ( n ) of positive dimensionsuch that for γ / ∈ Σ X,Y , X and γY satisfy Condition (1).Proof. We note first that { γT j } mj =0 is a Whitney stratification of γY . Let S = { } be a stratum of X and let T = { } be a stratum of Y . We have to prove that thereexists a definable subset Σ S,T ⊂ SO ( n ) of positive codimension such that S and γT intersect transversally in a neighborhood of 0. If dim S = n or dim T = n , there isnothing to prove so we can assume that e := dim S ≤ n − d := dim T ≤ n − M be the following definable set: M = (cid:8) ( p, γ ) ∈ ( U ′ \ { } ) × O ( n ) | p ∈ S ∩ γ − T (cid:9) , 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 , 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 ina neighborhood of p and f is a definable submersion such that T is locally definedby 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, whichimplies that Df ( γp ) | ( γp ) ⊥ : ( γp ) ⊥ → R n − d is a surjection. By Lemma 4.8, themapping 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 thesubmatrix 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 − n − n ( n + 1)2 + ( e + d ) = n ( n − e + d ) − n. Let π : M → O ( n ) be the natural projection. By Sard’s theorem (see [5]), itsdiscriminant ∆ 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 ). (cid:3) Lemma 4.11.
Assume that X and Y satisfy Condition (1). There exists a definablesubset Γ X,Y ⊂ S n − of positive codimension such that for v / ∈ Γ X,Y , v satisfiesCondition (2). Proof.
Let S be a stratum of X and let T be a stratum of Y . We have to provethat there exists a definable subset Γ S,T ⊂ S n − of positive codimension such that b S and c T v intersect transversally outside (0 , , S = { } and T = { } , then b S ∩ c T v = { (0 , } and there is nothing to prove.If dim S = n or dim T = n , there is nothing to prove neither. Let us treat first thecase 0 < e := dim S ≤ n − < d := dim T ≤ n −
1. Let M be the followingdefinable set: M = n ( p, τ, ν ) ∈ U × R × ( R n \ { } ) | ( p, τ ) ∈ b S ∩ c T ν o . Let us prove that M is a definable submanifold. Let ( p, τ, ν ) be a point in M . Thereis an open neighborhood V of ( p, τ, ν ) ∈ U × R × ( R n \ { } ) such that in V , M isdefined 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 submersionsuch that T is locally defined by f ( x ) = 0 in a neighborhood of p − τ ν . It is easy tocheck 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 definablesubmanifold of dimension 2 n + 1 − n + ( e + d ) = ( e + d ) + 1. Let π : M → R n bethe projection π ( p, τ, ν ) = ν . By Sard’s theorem (see [5]), its discriminant ∆ is adefinable subset of positive codimension. We take Γ S,T = S n − \ ∆. The remainingtwo cases are proved with the same method. (cid:3) The link of Z v ∩ { t ≥ } . We study the link of the set Z v ∩ { t ≥ } . We stillassume that X and Y satisfy Condition (1) and that v satisfies Condition (2). For( x, t ) ∈ R n +1 , we set ω ( x, t ) = | x | . Let ǫ > B nǫ ⊂ U . We setΓ Y = (cid:26) ( 1 u y, u ) | y ∈ Y, u ∈ ]0 , ǫ [ (cid:27) ⊂ R n +1 . We recall that the tangent cone of Y at 0 is C Y = Γ Y ∩ ( R n × { } ). It is a closedand conic definable set in R n . Lemma 4.12. If v / ∈ ( − C ( Y )) ∩ S n − then there exist ǫ v > and a > such thatthe inclusion c Y v ∩ { t ≥ } ∩ B n +1 ǫ v ⊂ (cid:8) ( x, t ) ∈ R n +1 | | x | ≥ at (cid:9) ∩ B n +1 ǫ v holds.Proof. If it is not the case then we can find a sequence of points ( x n , t n ) n ∈ N in c Y v \ { t = 0 } such that ( x n , t n ) → (0 ,
0) and lim n → + ∞ | x n | t n = 0. We have | x n − t n v | = | x n | + t n − t n h x n , v i , and so | x n − t n v | t n = | x n | t n + 1 − h x n t n , v i . Since |h x n t n , v i| ≤ | x n | t n , we find that lim n → + ∞ | x n − t n v | t n = 1. Since | x n || x n − t n v | = | x n | t n × t n | x n − t n v | , we find that lim n → + ∞ | x n || x n − t n v | = 0. Therefore we see thatlim n → + ∞ x n − t n v | x n − t n v | = − v, which implies that − v belongs to C ( Y ) ∩ S n − . (cid:3) RINCIPAL KINEMATIC FORMULAS FOR GERMS OF CLOSED DEFINABLE SETS 17
Corollary 4.13. If v / ∈ ( − C ( Y )) ∩ S n − then for < ǫ ≪ , the link of c Z v ∩{ t ≥ } is homemorphic to c Z v ∩ { t ≥ } ∩ { ( x, t ) | | x | = ǫ } .Proof. It is enough to show the implication (cid:26) ( y , t ) ∈ c Y v ∩ { t ≥ } ∩ B n +1 ǫ v ω ( y , t ) = 0 ⇒ ( y , t ) = (0 , . But if ω ( y , t ) = 0 then | y | = 0 and, by the previous lemma, | y | ≥ at for some a > (cid:3) If dim C ( Y ) ≤ n − v in S n − since dim C ( Y ) ∩ S n − < n − C ( Y ) = n . This implies thatdim Y = n . Let us denote by Y ′ the union of the strata of Y of dimension less thanor 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 ∈ S and that dim C S = n . If v is in C S \ C ( S \ S ) then there is α > such that ]0 , α ] · v ⊂ S .Proof. We note that C S \ C ( S \ S ) is not empty because dim C ( S ) = n . Let v ∈ C S \ C ( S \ S ) (note that necessarly v = 0). Let us suppose that for all α > , α ] · v is not included in S . Hence we can construct a sequence ( z n ) n ∈ N in thecomplement c S of S such that ( z n ) tends to 0 and z n | z n | = v . This implies that v belongs to C ( c S ). Since v belongs to C ( S ) as well, there exists a sequence ofpoints ( x n ) n ∈ N in S such that x n | x n | tends to v .Let a ∈ S and let b ∈ c S . Let [ a, b ] be the segment with extremities a and b , i.e.[ a, b ] = { z | z = λa + (1 − λ ) b, λ ∈ [0 , } . Since S is open, there exists 0 < η ≤ 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 . Actuallyif 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 .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 ) | tendsto 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 , y n | y n | = α n · v + β n · x n | x n | , with α n , β n ≥
0. Since h v, x n | x n | i →
1, there is n such that for n ≥ n , h v, x n | x n | i ≥ .This implies that for n ≥ n , 0 ≤ α n + β n + α n β n ≤ 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 ≥ β n tends to β ≥
0. Hence y n | y n | tendsto ( α + β ) v , where α + β = 1 for the limit of y n | y n | is a unit vector. We see that v belongs to C ( S \ S ), which is not possible by hypothesis. We conclude that thereis α > , α ] · v ⊂ S . (cid:3) Lemma 4.15.
Let W ⊂ R n be a closed definable set equipped with a Whitneystratification. Suppose that ∈ W and that lies in a stratum of dimension greaterthan or equal to 1. Let g : W → R be a definable function, restriction of a C definable function, such that is not a stratified critical point of g . Let f : W ∩{ g ≤ } → R be a definable function, restriction of a C definable function, such that f (0) = 0 and is a local strict maximum of f . Then ind( f, W ∩ { g ≤ } ,
0) = 0 .Proof.
By Lemma 3.3 in [18], χ (Lk( W ∩ { g ≤ } )) = 1. Since − f | W ∩{ g ≤ } is adistance function to the origin, we can writeind( f, W ∩ { g ≤ } ,
0) = 1 − χ ( W ∩ { g ≤ } ∩ {− f ≤ ǫ } ∩ { f = − δ } ) , with 0 < δ < ǫ ≪
1. But W ∩ { g ≤ } ∩ {− f ≤ ǫ } ∩ { f = − δ } is homeomorphic tothe link of W ∩ { g ≤ } at 0. (cid:3) Let us choose v in ( − C ( Y )) ∩ S n − such that v / ∈ ( − C ( Y ′ )) ∩ S n − . By Lemma4.12, there exists ǫ v > a > c Y ′ v ∩ { t ≥ } ∩ B n +1 ǫ v ⊂ { ( x, t ) | | x | ≥ at } ∩ B n +1 ǫ v . Lemma 4.16.
Under these assumptions, we have χ (cid:18) X ∩ ( Y + 2 ǫa v ) ∩ B nǫ (cid:19) = 1 , for < ǫ ≪ .Proof. For 0 < ǫ ≪
1, we set r ǫ = q ǫ + (cid:0) ǫa (cid:1) . The set X ∩ ( Y + ǫa ) ∩ B nǫ isthe intersection Z v ∩ { t = ǫa } ∩ B n +1 r ǫ . If ( x , t ) lies in Z v ∩ { t ≥ ǫa } ∩ B n +1 r ǫ then | x | ≤ ǫ + (cid:0) ǫa (cid:1) − t and | x | t ≤ ǫ t ≤ (cid:0) a (cid:1) . Therefore if ǫ is sufficiently small, Z v ∩ { t ≥ ǫa } ∩ B n +1 r ǫ ∩ c Y ′ v = ∅ .The function t | Z v has an isolated stratified critical value at 0. If ǫ is smallenough, the stratified critical points of t | Z v ∩{ t> ǫa }∩ B n +1 rǫ lie in Z v ∩ { t > ǫa } ∩ S nr ǫ .But c Y ′ v does not intersect Z v ∩ { t > ǫa } ∩ B n +1 r ǫ , so Z v ∩ { t > ǫa } ∩ ˚ B n +1 r ǫ (resp. Z v ∩ { t > ǫa } ∩ S nr ǫ ) is stratified by strata of the form b S ∩ c T v ∩ { t > ǫa } ∩ ˚ B n +1 r ǫ (resp. b S ∩ c T v ∩ { t > ǫa } ∩ S nr ǫ ), where S is a stratum of X and T is a stratum of Y of dimension n . This means that Z v ∩ { t > ǫa } ∩ ˚ B n +1 r ǫ (resp. Z v ∩ { t > ǫa } ∩ S nr ǫ )is stratified by open subsets of strata of the form b S ∩ { t > ǫa } ∩ ˚ B n +1 r ǫ (resp. b S ∩ { t > ǫa } ∩ S nr ǫ ).Such a stratum b S is a product S × ] − ǫ ′ , ǫ ”[ where ǫ ′ , ǫ ” > S is astratum of X . A point ( x , t ) is a critical point of t | b S ∩ S nrǫ if and only if x ∈ S andrank N ( x ) 0... ... N c s ( x ) 0 x t < c S + 2 , RINCIPAL KINEMATIC FORMULAS FOR GERMS OF CLOSED DEFINABLE SETS 19 where ( N ( x ) , . . . , N c S ( x )) is a basis of the normal space to S at x , that is if andonly if rank N ( x )... N c s ( x ) x < c S + 1 . But if x is a point of S close to 0 but distinct from 0, the sphere S n − | x | intersectsthe stratum S transversally. We conclude that the unique possible critical point of t | b S ∩ S nrǫ is the point (0 , r ǫ ). But v is in ( − C Y \ − C Y ′ ) ∩ S n − , so by Lemma 4.14,there is α > , α ] · ( − v ) ⊂ T where T is a stratum of Y of dimension n . Hence { }× ]0 , α ] is included in c T v . We conclude that for 0 < ǫ ≪
1, (0 , r ǫ ) isthe only critical point of t | Z v ∩{ t> ǫa }∩ B n +1 rǫ . Moreover it is a strict local maximum.Applying Theorem 3.1 in [13] and Lemma 4.15, we get χ (cid:18) Z v ∩ { t ≥ ǫa } ∩ B n +1 r ǫ (cid:19) − χ (cid:18) Z v ∩ { t = 2 ǫa } ∩ B n +1 r ǫ (cid:19) = 0 , and χ (cid:18) Z v ∩ { t ≥ ǫa } ∩ B n +1 r ǫ (cid:19) = ind( − t, Z v ∩ B n +1 r ǫ , (0 , r ǫ )) = 1 , because − t | Z v ∩ B n +1 rǫ has a strict local minimum at the point (0 , r ǫ ). (cid:3) Corollary 4.17.
Under the same assumptions, we have χ (Lk( Z v ∩ { t ≥ } )) = χ c (cid:18) Z v ∩ { t ≥ } ∩ { ( x, t ) | | x | = ǫ, t < ǫa } (cid:19) + 1 . Proof.
Let h ( x, t ) be the semi-algebraic function defined by h ( x, t ) = max (cid:16) | x | , a t (cid:17) . As explained by Durfee in [11], Section 3, the link of Z v ∩ { t ≥ } is homeomorphicto Z v ∩ { t ≥ } ∩ { h = ǫ } for 0 < ǫ ≪
1. We have Z v ∩ { t ≥ } ∩ { h = ǫ } = Z v ∩ { t ≥ } ∩ {| x | = ǫ, at ≤ ǫ } [ Z v ∩ { t ≥ } ∩ {| x | ≤ ǫ, at ǫ } . It is enough to use the previous lemma and the additivity of χ c . (cid:3) On the two sides of the kinematic formulas.
We prove the existenceof the left-hand sides of the kinematic formulas, and we show that both sides ofthe formulas are symmetric in X and Y . We also give a relation with the polarinvariants.Let ( X, ⊂ ( R n ,
0) and ( Y, ⊂ ( R n ,
0) be two germs of closed definable sets.We assume that X and Y are included in an open set U . Let ǫ > B nǫ ⊂ U .1) Let us fix ( ǫ, δ ) such that 0 ≤ ǫ ≤ ǫ and 0 ≤ δ ≤ ǫ . Let A = (cid:8) ( x, γ, v ) ∈ R n × SO ( n ) × S n − | x ∈ X, x − δv ∈ γY, | x | ≤ ǫ (cid:9) . It is a closed definable set. By Hardt’s theorem applied to the projection π : A → SO ( n ) × S n − , the function ( γ, v ) χ ( X ∩ ( γY + δv ) ∩ B nǫ ) takes a finite number of values. As in [3] we equip SO ( n ) with the Haar measure dγ , normalized in such away that the volume of SO ( n ) is s n − . We equip S n − with the usual Riemanniannmeasure (or density) dv and SO ( n ) × S n − with the product measure dγdv . Withthis measure, the function ( γ, v ) χ ( X ∩ ( γY + δv ) ∩ B nǫ ) is integrable and so theintegral Z SO ( n ) × S n − χ ( X ∩ ( γY + δv ) ∩ B nǫ ) dγdv exists and is finite. Moreover the function γ Z S n − χ ( X ∩ ( γY + δv ) ∩ B nǫ ) dv is integrable and the function v Z SO ( n ) χ ( X ∩ ( γY + δv ) ∩ B nǫ ) dγ is integrable and we have Z SO ( n ) × S n − χ ( X ∩ ( γY + δv ) ∩ B nǫ ) dγdv = Z SO ( n ) [ Z S n − χ ( X ∩ ( γY + δv ) ∩ B nǫ ) dv ] dγ = Z S n − [ Z SO ( n ) χ ( X ∩ ( γY + δv ) ∩ B nǫ ) dγ ] dv. Now let us fix ǫ > < ǫ ≤ ǫ . By Hardt’s theorem, for every( γ, v ) ∈ SO ( n ) × S n − , there is a small interval ]0 , δ ǫ [ such that the function δ χ ( X ∩ ( γY + δv ) ∩ B nǫ ) is constant on ]0 , δ ǫ [ and so lim δ → + χ ( X ∩ ( γY + δv ) ∩ B nǫ )exists and is this constant value. Similarly as above, the function ( γ, v, δ ) χ ( X ∩ ( γY + δv ) ∩ B nǫ ) takes a finite number of values and so, by Lebesgue’s the-orem, the function ( γ, v ) lim δ → + χ ( X ∩ ( γY + δv ) ∩ B nǫ ) is integrable and wehave lim δ → + Z SO ( n ) × S n − χ ( X ∩ ( γY + δv ) ∩ B nǫ ) dγdv = Z SO ( n ) × S n − lim δ → + χ ( X ∩ ( γY + δv ) ∩ B nǫ ) dγdv. ( ∗ )Let us fix ( γ, v ) ∈ SO ( n ) × S n − and let B = n ( x, ǫ, δ ) ∈ R n × { ( ǫ, δ ) | < ǫ ≤ ǫ , < δ ≤ ǫ } | x ∈ X, x − δv ∈ Y, | x | ≤ ǫ o . It is a closed definable set. Applying the argument of the proof of Lemma 3.1, wesee that there exists 0 < ǫ ≤ ǫ and a definable function r :]0 , ǫ ] → R continuous,monotone and strictly positive such that the function( ǫ ′ , δ ′ ) χ ( X ∩ ( γY + δv ) ∩ B nǫ ′ )is constant on { ( ǫ ′ , δ ′ ) | < ǫ ′ < ǫ , < δ ′ < r ( ǫ ′ ) } . But we see that for ǫ ′ ∈ ]0 , ǫ [and 0 < δ ′ < r ( ǫ ′ ), χ ( X ∩ ( γY + δ ′ v ) ∩ B nǫ ′ ) = lim δ → + χ ( X ∩ ( γY + δv ) ∩ B nǫ ′ ) . Therefore the limit lim ǫ → lim δ → + χ ( X ∩ ( γY + δv ) ∩ B nǫ ) exists and equals theabove constant value. Always by Hardt’s theorem, the function ( γ, v, ǫ, δ ) RINCIPAL KINEMATIC FORMULAS FOR GERMS OF CLOSED DEFINABLE SETS 21 χ ( X ∩ ( γY + δv ) ∩ B nǫ ) takes a finite number of values and so does the function( γ, v, ǫ ) lim δ → + χ ( X ∩ ( γY + δv ) ∩ B nǫ ). By Lebesgue’s theorem, the function( γ, v ) lim ǫ → lim δ → + χ ( X ∩ ( γY + δv ) ∩ B nǫ ) is integrable andlim ǫ → Z SO ( n ) × S n − lim δ → + χ ( X ∩ ( γY + δv ) ∩ B nǫ ) dγdv = Z SO ( n ) × S n − lim ǫ → lim δ → + χ ( X ∩ ( γY + δv ) ∩ B nǫ ) dγdv. Finally, by Equality ( ∗ ) above, we have thatlim ǫ → [ lim δ → + Z SO ( n ) × S n − χ ( X ∩ ( γY + δv ) ∩ B nǫ ) dγdv ]exists and equals Z SO ( n ) × S n − lim ǫ → lim δ → + χ ( X ∩ ( γY + δv ) ∩ B nǫ ) dγdv. Definition 4.18.
For two germs of closed definable sets ( X, ⊂ ( R n ,
0) and( Y, ⊂ ( R n , σ ( X, Y,
0) = 1 s n − Z SO ( n ) × S n − lim ǫ → lim δ → + χ ( X ∩ ( γY + δv ) ∩ B nǫ ) dγdv.
2) Let us fix ǫ such that 0 ≤ ǫ ≤ ǫ . Let C = (cid:8) ( x, γ, H ) ∈ R n × SO ( n ) × G n − n | x ∈ X ∩ γY ∩ H, | x | = ǫ (cid:9) . It is a closed definable set. By Hardt’s theorem applied to the projection π : C → SO ( n ) × G nn − , the function ( γ, H ) χ (cid:0) X ∩ γY ∩ H ∩ S n − ǫ (cid:1) takes a finite numberof 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 − n with the SO ( n )-invariant mea-sure (or density) dH and SO ( n ) × G n − n with the product measure. With thismeasure, the function ( γ, H ) χ (Lk( X ∩ γY ∩ H )) is integrable and so the in-tegral R SO ( n ) × G n − n χ (Lk( X ∩ γY ∩ H )) dγdH exists. Moreover the function γ R G n − n χ (Lk( X ∩ γY ∩ H )) dH is integrable and so is the function( γ, v ) Z G n − n χ (Lk( X ∩ γY ∩ H )) dH on SO ( n ) × S n − . 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 − .3) By Lemmas 4.10 and 4.11, there exists a definable subset ∆ ⊂ SO ( n ) × S n − of positive codimension such that for ( γ, v ) / ∈ ∆,lim ǫ → lim δ → + Λ ( X ∩ ( γY + δv ) , X ∩ ( γY + δv ) ∩ B nǫ ) =lim ǫ → lim δ → + χ ( X ∩ ( γY + δv ) ∩ B nǫ ) − χ (Lk( X ∩ γY )) − s n − Z S n − χ (Lk( X ∩ γY ∩ { u ∗ = 0 } )) du. Therefore the function( γ, v ) lim ǫ → lim δ → + Λ ( X ∩ ( γY + δv ) , X ∩ ( γY + δv ) ∩ B nǫ ) is integrable on SO ( n ) × S n − . Definition 4.19.
For two germs of closed definable set ( X, ⊂ ( R n ,
0) and( Y, ⊂ ( R n , lim0 ( X, Y, s n − Z SO ( n ) × S n − lim ǫ → lim δ → + Λ ( X ∩ ( γY + δv ) , X ∩ ( γY + δv ) ∩ B nǫ ) dγdv. We note thatΛ lim0 ( X, Y,
0) = σ ( X, Y, − s n − Z SO ( n ) χ (Lk( X ∩ γY )) dγ − s n − Z SO ( n ) Z S n − χ (Lk( X ∩ γY ∩ { u ∗ = 0 } )) dudγ. The two limits σ ( X, Y,
0) and Λ lim0 ( X, Y,
0) are symmetric in X and Y , as explainedin the next proposition. Proposition 4.20.
For two germs of closed definable sets ( X, ⊂ ( R n , and ( Y, ⊂ ( R n , , we have σ ( X, Y,
0) = σ ( Y, X, and Λ lim0 ( X, Y,
0) = Λ lim0 ( Y, X, . Proof.
By Lemma 4.6, we know thatlim ǫ → lim δ → + χ ( X ∩ ( γY + δv ) ∩ B nǫ ) = lim ǫ → lim δ → + χ (( X − δv ) ∩ γY ∩ B nǫ ) . The change of variables v
7→ − v gives that Z S n − χ (( X − δv ) ∩ γY ∩ B nǫ ) dv = Z S n − χ (( X + δv ) ∩ γY ∩ B nǫ ) dv, and so Z S n − χ (( X − δv ) ∩ γY ∩ B nǫ ) dv = Z S n − χ (cid:0) γ − ( X + δv ) ∩ Y ∩ B nǫ (cid:1) dv. Hence, by the change of variables γ γ − on SO ( n ), we are lead to compute Z SO ( n ) [ Z S n − χ ( γ ( X + δv ) ∩ Y ∩ B nǫ ) dv ] dγ. But for γ ∈ SO ( n ), the change of variables u γu gives that Z S n − χ ( γ ( X + δu ) ∩ Y ∩ B nǫ ) du = Z S n − χ (( γX + δv ) ∩ Y ∩ B nǫ ) dv. Finally we get that Z SO ( n ) Z S n − χ ( γ ( X + δv ) ∩ Y ∩ B nǫ ) dvdγ = Z SO ( n ) Z S n − χ (( γX + δv ) ∩ Y ∩ B nǫ ) dvdγ. It is enough to pass to the limits to get the equality σ ( X, Y,
0) = σ ( Y, X, lim0 ( X, Y,
0) is obtained applying the relation between σ ( X, Y,
0) andΛ lim0 ( X, Y, (cid:3)
Now let us relate σ ( X, Y,
0) with the polar invariants of Comte and Merle [9].
RINCIPAL KINEMATIC FORMULAS FOR GERMS OF CLOSED DEFINABLE SETS 23
Proposition 4.21. If H ∈ G n − kn , k ∈ { , . . . , n } , then we have σ ( X, H,
0) = σ k ( X, . Proof.
The proof is straightforward for k = 0, because in this case σ ( X, R n ,
0) =lim ǫ → χ ( X ∩ B nǫ ) = 1.Let us assume that k >
0. First we note that σ ( X, H,
0) = 1 g n − kn Z G n − kn (cid:18) s n − Z S n − lim ǫ → lim δ → + χ ( X ∩ ( H + δv ) ∩ B nǫ ) dv (cid:19) dH. For H ∈ G n − kn , we recall that S k − H ⊥ is the unit sphere in H ⊥ and we denote by p H ⊥ the orthogonal projection onto H ⊥ . If v ∈ S k − H ⊥ and w ∈ S n − are such that p H ⊥ ( w ) | p H ⊥ ( w ) | = v , thenlim ǫ → lim δ → + χ ( X ∩ ( H + δv ) ∩ B nǫ ) = lim ǫ → lim δ → + χ ( X ∩ ( H + δw ) ∩ B nǫ ) . This implies that σ ( X, H,
0) = 1 g n − kn Z G n − kn s k − Z S k − H ⊥ lim ǫ → lim δ → + χ ( X ∩ ( H + δv ) ∩ B nǫ ) dv ! dH, (see the proof of Corollary 5.7 in [12] for a similar argument or use the co-areaformula). To end the proof, it is enough to show, with the notations of Section 3,that N P X i =1 χ Pi · Θ( K pi ,
0) = 1 s k − Z S k − P lim ǫ → lim δ → + χ (cid:0) X ∩ ( P ⊥ + δv ) ∩ B nǫ (cid:1) dv. By Lemma 4.14, we know that if v ∈ C K Pi \ C ( K Pi \ K Pi ), there is δ > , δ ] · v ⊂ K Pi . Hence1 s k − Z S k − P lim ǫ → lim δ → + χ (cid:0) X ∩ ( P ⊥ + δv ) ∩ B nǫ (cid:1) dv = N P X i =1 χ Pi · vol( C K Pi ∩ S k − P ) s k − . By [26] Lemma 2.1, vol( C K Pi ∩ S k − P ) s k − is exactly Θ k ( K Pi , (cid:3) Remark 4.22.
The equality σ k ( X,
0) = 1 g n − kn Z G n − kn s k − Z S k − H ⊥ lim ǫ → lim δ → + χ ( X ∩ ( H + δv ) ∩ B nǫ ) dv ! dH, is natural because the two sides of the equality coincide for they measure the samemean-value of Euler characteristics. It already appeared in [9] page 244 in the coniccase, and we used it in [15, 16, 17] as a definition for the polar invariants. We proveit here for completeness.We end this subsection with another symmetry result. Lemma 4.23.
For two germs of closed definable sets ( X, ⊂ ( R n , and ( Y, ⊂ ( R n , , we have n X i =0 Λ lim i ( X, · σ n − i ( Y,
0) = n X i =0 σ i ( X, · Λ lim n − i ( Y, . Proof.
By Theorem 3.8, we get n X i =0 Λ lim i ( X, · σ n − i ( Y,
0) = n − X i =0 ( σ i ( X, − σ i +1 ( X, · σ n − i ( Y, σ n ( X, · σ ( Y,
0) = n − X i =0 σ i ( X, · σ n − i ( Y, − n − X i =0 σ i +1 ( X, · σ n − i ( Y,
0) + σ n ( X, · σ ( Y, . Therefore we obtain n X i =0 Λ lim i ( X, · σ n − i ( Y,
0) = σ ( X, · σ n ( Y, n − X i =1 σ i ( X, · ( σ n − i ( Y, − σ n − i +1 ( Y, σ n ( X, · ( σ ( Y, − σ ( Y, (cid:3) A new spherical kinematic formula
We give a new spherical kinematic formula for two definable subsets of the unitsphere.Let X ⊂ S n − be a compact definable set and let Y ⊂ S n − be a definableset, not necessarly compact. We recall that the ˜Λ i ’s, i = 0 , . . . , n −
1, denote thespherical Lipschitz-Killing measures.
Proposition 5.1.
The following kinematic formula holds: s n − Z SO ( n ) χ c ( X ∩ γY ) dγ = n − X i =0 ˜Λ i ( X, X ) s i · g i +1 n Z G i +1 n χ c ( Y ∩ H ) dH. Proof.
First step:
We study the case Y compact. Applying the generalized spher-ical Gauss-Bonnet formula (see Theorem 1.2 in [3]) to χ ( X ∩ γY ), we obtain Z SO ( n ) χ ( X ∩ γY ) dγ = n − X i =0 , ,... s i Z SO ( n ) ˜Λ i ( X ∩ γY, X ∩ γY ) dγ. Then we apply the generalized spherical kinematic formula (see [3, 21]) to each˜Λ i ( X ∩ γY, X ∩ γY ) and we get Z SO ( n ) χ ( X ∩ γY ) dγ = n − X i =0 , ,... s i X p + q = i + n − s i s n − s p s q ˜Λ p ( X, X )˜Λ q ( Y, Y ) . Therefore we have1 s n − Z SO ( n ) χ ( X ∩ γY ) dγ = n − X i =0 , ,... X p + q = i + n − ˜Λ p ( X, X ) s p q ( Y, Y ) s q = X p + q = n − ˜Λ p ( X, X ) s p q ( Y, Y ) s q + X p + q = n +1 ˜Λ p ( X, X ) s p q ( Y, Y ) s q RINCIPAL KINEMATIC FORMULAS FOR GERMS OF CLOSED DEFINABLE SETS 25 + · · · + X p + q = e ( n ) ˜Λ p ( X, X ) s p q ( Y, Y ) s q , where e ( n ) = 2 n − n − n − n − s n − Z SO ( n ) χ ( X ∩ γY ) dγ = ˜Λ n − ( X, X ) s n − X q =0 , ,... q ( Y, Y ) s q ! + ˜Λ n − ( X, X ) s n − X q =1 , ,... q ( Y, Y ) s q ! + · · · + ˜Λ ( X, X )2 n − ( Y, Y ) s n − ! . In [14] pages 175-176, we proved that1 g n Z G n χ ( Y ∩ H ) dH = 2 ˜Λ n − ( Y, Y ) s n − , g n Z G n χ ( Y ∩ H ) dH = 2 ˜Λ n − ( Y, Y ) s n − , and for k ≥ g k − n Z G k − n χ ( Y ∩ H ) dH = X i =2 , ,... n − k + i ( Y, Y ) s n − k + i = X q = n − k +2 ,n − k +4 ,... q ( Y, Y ) s q . Applying these relations, we get that1 s n − Z SO ( n ) χ ( X ∩ γY ) dγ = ˜Λ n − ( X, X ) s n − χ ( Y )+ ˜Λ n − ( X, X ) s n − g n − n Z G n − n χ ( Y ∩ H ) dH + ˜Λ n − ( X, X ) s n − g n − n Z G n − n χ ( Y ∩ H ) dH + · · · + ˜Λ ( X, X ) s g n Z G n χ ( Y ∩ H ) dH. Second step:
Let Y ⊂ S n − be compact and let K ( Y be a compact definableset. By the first step, we have1 s n − Z SO ( n ) χ ( X ∩ γY ) dγ = n − X i =0 ˜Λ( X, X ) s i · g i +1 n Z G i +1 n χ ( Y ∩ H ) dH, and 1 s n − Z SO ( n ) χ ( X ∩ γK ) dγ = n − X i =0 ˜Λ( X, X ) s i · g i +1 n Z 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 ) and1 s n − Z SO ( n ) χ c ( X ∩ γ ( Y \ K )) dγ = n − X i =0 ˜Λ i ( X, X ) s i · g i +1 n Z G i +1 n [ χ ( Y ∩ H ) − χ ( K ∩ H )] dH = n − X i =0 ˜Λ i ( X, X ) s i · g i +1 n Z 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 thefollowing cell decomposition Y = ⊔ rj =1 C j , where each C j is a definable subsethomeomorphic to a unit cube ]0 , d j . By the second step, the result is valid foreach cell C j , because C j and C j \ C j are compact and definable. By additivity of χ c , we have1 s n − Z SO ( n ) χ c ( X ∩ γY ) dγ = 1 s n − Z SO ( n ) χ c (cid:0) X ∩ γ ( ⊔ rj =1 C j ) (cid:1) dγ = 1 s n − Z SO ( n ) χ c (cid:0) X ∩ ( ⊔ rj =1 γC j ) (cid:1) dγ = 1 s n − Z SO ( n ) χ c (cid:0) ⊔ rj =1 ( X ∩ γC j ) (cid:1) dγ = r X j =1 s n − Z SO ( n ) χ c ( X ∩ γC j ) dγ. Applying the second step, we obtain1 s n − Z SO ( n ) χ c ( X ∩ γY ) dγ = r X j =1 n − X i =0 ˜Λ i ( X, X ) s i · g i +1 n Z G i +1 n χ c ( C j ∩ H ) dH = n − X i =0 ˜Λ i ( X, X ) s i · g i +1 n Z G i +1 n r X j =1 χ c ( C j ∩ H ) dH = n − X i =0 ˜Λ i ( X, X ) s i · g i +1 n Z G i +1 n χ c ( Y ∩ H ) dH. (cid:3) A second kinematic formula in the unit ball
We deduce from the previous spherical kinematic a new kinematic formula fordefinable subsets of the unit ball.Let X ⊂ R n be a closed conic definable set. Let Y ⊂ B n be another definableset. Proposition 6.1.
The following kinematic formula holds: s n − Z SO ( n ) χ c ( X ∩ γY ) dγ = n X i =0 Λ i ( X, X ∩ B n ) b i · g in Z G in χ c ( Y ∩ H ) dH. Proof.
Let us assume first that 0 / ∈ Y and let φ be the following definable mapping: φ : B n \ { } → S n − x x | x | . By Hardt’s theorem, there exists a definable partition of φ ( Y ), φ ( Y ) = ⊔ rj =1 W j ,such that for j ∈ { , . . . , r } , the mapping φ | Y ∩ φ − ( W j ) : φ − ( W j ) ∩ Y → W j is trivial.By additivity and multiplicity of χ c , we can write χ c ( Y ) = P rj =1 α j χ c ( W j ), where α j = χ c ( F j ) with F j the fibre of φ | Y ∩ φ − ( W j ) . Let us set X ∗ = ( X ∩ B n ) \ { } .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 ∗ ) = ⊔ rj =1 W j ∩ φ ( X ∗ ). Note that if w ∈ φ ( X ∗ ) then φ − ( w ) ⊂ X ∗ by theconic structure of X . Hence if w j ∈ W j ∩ φ ( X ∗ ), φ − ( w j ) ∩ Y ∩ X = φ − ( w j ) ∩ Y RINCIPAL KINEMATIC FORMULAS FOR GERMS OF CLOSED DEFINABLE SETS 27 and χ c ( φ − ( 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 X 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 ) = ⊔ rj =1 γW j and for j ∈ { , . . . , r } , the mapping φ | γY ∩ φ − ( γW j ) : γY ∩ φ − ( γW j ) → γW j is trivial, with fibre homeomorphic to F j . As above, we can write χ c ( X ∩ γY ) = r X 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 get1 s n − Z SO ( n ) χ c ( X ∩ γY ) dγ = r X j =1 α j s n − Z SO ( n ) χ c ( γW j ∩ φ ( X ∗ )) dγ = r X j =1 α j n − X i =1 ˜Λ i ( φ ( X ∗ ) , φ ( X ∗ )) s i · g i +1 n Z G i +1 n χ c ( W j ∩ H ) dH ! = n − X i =1 ˜Λ i ( φ ( X ∗ ) , φ ( X ∗ )) s i · g i +1 n Z G i +1 n r X j =1 α j χ c ( W j ∩ H ) dH. Since H is conic, P rj =1 α j χ c ( W j ∩ H ) = χ c ( Y ∩ H ) by the above argument. ApplyingCorollary 3.5 in [14], we obtain the following equality:1 s n − Z SO ( n ) χ c ( X ∩ γY ) dγ = n − X i =0 Λ i +1 ( X, X ∩ B n ) b i +1 · g i +1 n Z G i +1 n χ c ( Y ∩ H ) dH = n X i =1 Λ i ( X, X ∩ B n ) b i · g in Z G in 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 ) + 1where Y ∗ = Y \ { } . Therefore we have1 s n − Z SO ( n ) χ c ( X ∩ γY ) dγ = 1 + 1 s n − Z SO ( n ) χ c ( X ∩ γY ∗ ) dγ = 1 + n X i =1 Λ i ( X, X ∩ B n ) b i · g in Z G in χ c ( Y ∗ ∩ H ) dH = 1 + n X i =1 Λ i ( X, X ∩ B n ) b i · g in Z G in χ c ( Y ∩ H ) dH − n X i =1 Λ i ( X, X ∩ B n ) b i . But by Corollary 5.2 in [14], we know that 1 − P ni =1 Λ i ( X,X ∩ B n ) b i = Λ ( X, X ∩ B n ) . (cid:3) A infinitesimal kinematic formula for conic sets
We prove a first version of the infinitesimal principal kinematic formula for closedconic definable sets.Let
X, Y ⊂ R n be two closed conic definable sets. We keep the notations usedin Section 4. Lemma 7.1.
There exists δ > such that for < δ ≤ δ , the topological types of Z v,δ ∩ B n does not depend on the choice δ . Moreover, we have lim δ → χ ( Z v,δ ∩ B n ) = lim ǫ → lim δ → χ ( Z v,δ ∩ B nǫ ) . Proof.
By Lemma 4.4, we know that there exists ǫ > < ǫ ≤ ǫ ,there exits δ ǫ such that for 0 < δ ≤ δ ǫ , the topological type of Z v,δ ∩ B nǫ doesnot depend on the choice of the couple ( ǫ, δ ). Let us fix such a couple ( ǫ, δ ). Let θ ǫ : R n → R n be the diffeomorphism θ ǫ ( x ) = ǫ x . Then θ ǫ ( Z v,δ ∩ B nǫ ) = Z v, δǫ ∩ B n .Since lim δ → δǫ = 0, we get the result. (cid:3) We are in position to state a first infinitesimal kinematic formula in the conicsetting.
Proposition 7.2.
Let
X, Y ⊂ R n be two closed conic definable sets. The followingkinematic formula holds: s n − Z SO ( n ) × S n − lim δ → + χ ( X ∩ ( γY + δv ) ∩ B n ) dγdv = n X i =0 Λ i ( X, X ∩ B n ) b i · σ n − i ( Y, . Proof.
Let us fix δ >
0. By the change of variable u = γv , we have that for γ ∈ SO ( n ) Z S n − χ ( X ∩ ( γY + δu ) ∩ B n ) du = Z S n − χ ( X ∩ γ ( Y + δv ) ∩ B n ) dv. Applying Proposition 6.1 to X ∩ B n and ( Y + δv ) ∩ B n , we get that1 s n − Z SO ( n ) × S n − χ ( X ∩ γ ( Y + δv ) ∩ B n ) dγdv = 1 s n − Z S n − Z SO ( n ) χ ( X ∩ γ ( Y + δv ) ∩ B n ) dγdv = n X i =1 Λ i ( X, X ∩ B n ) b i · g in s n − Z S n − Z G in χ (( Y + δv ) ∩ B n ∩ H ) dHdv, and so that1 s n − Z SO ( n ) × S n − χ ( X ∩ γ ( Y + δv ) ∩ B n ) dvdγ = n X i =1 Λ i ( X, X ∩ B n ) b i · g in Z G in s n − Z S n − χ (( Y + δv ) ∩ B n ∩ H ) dvdH. Passing to the limit as δ → + and using Lebesgue’s theorem, we obtain that1 s n − Z SO ( n ) × S n − lim δ → + χ ( X ∩ ( γY + δv ) ∩ B n ) dvdγ RINCIPAL KINEMATIC FORMULAS FOR GERMS OF CLOSED DEFINABLE SETS 29 = n X i =1 Λ i ( X, X ∩ B n ) b i · g in Z G in s n − Z S n − lim δ → + χ (( Y + δv ) ∩ B n ∩ H ) dvdH. By Lemmas 4.6 and 7.1 applied to Y and H , we have thatlim δ → + χ (( Y + δv ) ∩ B n ∩ H ) = lim δ → + χ ( Y ∩ B n ∩ ( H − δv )) , and so 1 g in Z G in s n − Z S n − lim δ → + χ (( Y + δv ) ∩ B n ∩ H ) dvdH = σ n − i ( Y, , by Proposition 4.21. (cid:3) The principal kinematic formulas
We prove our main results : the principal kinematic formulas for germs of closeddefinable sets. We will use the kinematic formula for closed conic definable setsproved in the previous section. We will proceed in several steps.We keep the notations used in Section 4. For convenience, we also use thenotation ω ( x ) for | x | , if x is in R n . First step: ( X, ⊂ ( R n ,
0) is a germ of closed definable set, Y ⊂ R n is a closedconic definable set.We assume that X is included in an open neighborhood U of 0. Let ǫ > B nǫ ⊂ U . For 0 < u ≤ ǫ , we set X u = X ∩ S n − u and we denote by CX u the cone over X u , i.e.: CX u = (cid:8) x ∈ R n | ∃ λ ∈ R + and z ∈ X u such that x = λz (cid:9) . Lemma 8.1.
There exists a definable subset ∆ Y ⊂ S n − of positive codimensionsuch that for v / ∈ ∆ Y , lim u → χ (cid:16) Lk( [ CX u ∩ c Y v ∩ { t ≥ } ) (cid:17) = χ (cid:16) Lk( b X ∩ c Y v ∩ { t ≥ } ) (cid:17) . Proof. If v / ∈ ( − Y ) ∩ S n − , then by Corollary 4.13, we have χ (cid:16) Lk( [ CX u ∩ c Y v ∩ { t ≥ } ) (cid:17) = lim ǫ → χ (cid:16) [ CX u ∩ c Y v ∩ { t ≥ } ∩ { ω = ǫ } (cid:17) and χ (cid:16) Lk( b X ∩ c Y v ∩ { t ≥ } ) (cid:17) = lim ǫ → χ (cid:16) b X ∩ c Y v ∩ { t ≥ } ∩ { ω = ǫ } (cid:17) . Let us choose ǫ ≥ < u ≤ ǫ , χ (cid:16) Lk( b X ∩ c Y v ∩ { t ≥ } ) (cid:17) = χ (cid:16) b X ∩ c Y v ∩ { t ≥ } ∩ { ω = u } (cid:17) . But b X ∩ { ω = u } = [ CX u ∩ { ω = u } and so χ (cid:16) b X ∩ c Y v ∩ { t ≥ } ∩ { ω = u } (cid:17) = χ (cid:16) [ CX u ∩ c Y v ∩ { t ≥ } ∩ { ω = u } (cid:17) . For any ǫ >
0, the mapping [ CX u ∩ c Y v ∩ { t ≥ } ∩ { ω = ǫ } → [ CX u ∩ c Y v ∩ { t ≥ } ∩ { ω = u } ( x, t ) ( uǫ x, uǫ t )is a homemorphism, since [ CX u ∩ c Y v ∩ { t ≥ } is conic and ω ( λ ( x, t )) = λω ( x, t ) forany λ >
0. Therefore χ (cid:16) Lk( [ CX u ∩ c Y v ∩ { t ≥ } ) (cid:17) = χ (cid:16) [ CX u ∩ c Y v ∩ { t ≥ } ∩ { ω = u } (cid:17) . If dim Y ≤ n − Y = S n − ∩ ( − Y ).If dim Y = n then let Y ′ be the union of the strata of Y of dimension less thanor equal to n −
1. If v ∈ ( − Y ) \ ( − Y ′ ), then by Corollary 4.17, we have χ (cid:16) Lk( [ CX u ∩ c Y v ∩ { t ≥ } ) (cid:17) = 1 + χ c (cid:18) [ CX u ∩ c Y v ∩ { t ≥ } ∩ { ω = ǫ, t < a ǫ } (cid:19) ,χ (cid:16) Lk( b X ∩ c Y v ∩ { t ≥ } ) (cid:17) = 1 + χ c (cid:18) b X ∩ c Y v ∩ { t ≥ } ∩ { ω = ǫ, t < a ǫ } (cid:19) , for 0 < ǫ ≪ a is such that c Y ′ v ∩ { t ≥ } ⊂ { ( x, t ) | ω ( x ) ≥ at } in aneighborhood of (0 , ǫ ≥ < u ≤ ǫ , χ (cid:16) Lk( b X ∩ c Y v ∩ { t ≥ } ) (cid:17) = 1 + χ c (cid:18) b X ∩ c Y v ∩ { t ≥ } ∩ { ω = u, t < a u } (cid:19) , and χ c (cid:18) b X ∩ c Y v ∩ { t ≥ } ∩ { ω = u, t < a u } (cid:19) = χ c (cid:18) [ CX u ∩ c Y v ∩ { t ≥ } ∩ { ω = u, t < a u } (cid:19) . Using the same homeomorphism as above, we can conclude that χ (cid:16) Lk( [ CX u ∩ c Y v ∩ { t ≥ } ) (cid:17) = 1 + χ c (cid:18) [ CX u ∩ c Y v ∩ { t ≥ } ∩ { ω = u, t < a u } (cid:19) . So if dim Y = n , we put ∆ Y = S n − ∩ ( − Y ′ ). (cid:3) Proposition 8.2. If Y ⊂ R n is a closed conic definable set, then for any germof closed definable set ( X, ⊂ ( R n , , the following principal kinematic formulaholds: σ ( X, Y,
0) = n X i =0 Λ lim i ( X, Y, · σ n − i ( Y, . Proof.
By Lemma 4.10, there exists a definable subset Σ
X,Y ⊂ SO ( n ) of positivecodimension such that for γ / ∈ Σ X,Y , X and γY satisfy Condition (1). Let usfix γ / ∈ Σ X,Y . By Lemma 4.11, there exists a definable subset Γ
X,γY ⊂ S n − ofpositive codimension such that for v / ∈ Γ X,γY , v satisfies Condition (2). Let uschoose v / ∈ Γ X,γY . By Lemma 4.2, the function t : b X ∩ \ ( γY ) v → R has an isolatedstratified critical point at (0 , ǫ → lim δ → + χ ( X ∩ ( γY + δv ) ∩ B nǫ ) = χ (cid:16) Lk( b X ∩ \ ( γY ) v ∩ { t ≥ } ) (cid:17) , and σ ( X, Y,
0) = 1 s n − Z SO ( n ) × S n − χ (cid:16) Lk( b X ∩ \ ( γY ) v ∩ { t ≥ } ) (cid:17) dγdv. Of course the same equality is true if we replace X with CX u . By Proposition 7.2for 0 < u ≤ ǫ , we have σ ( CX u , Y,
0) = n X i =0 Λ i ( CX u , CX u ∩ B n ) b i · σ n − i ( Y, . RINCIPAL KINEMATIC FORMULAS FOR GERMS OF CLOSED DEFINABLE SETS 31
By Lemma 8.1, for γ ∈ SO ( n ) and v / ∈ ∆ γY ,lim u → χ (cid:16) Lk( [ CX u ∩ c Y v ∩ { t ≥ } ) (cid:17) = χ (cid:16) Lk( b X ∩ c Y v ∩ { t ≥ } ) (cid:17) . Hence, by Hardt’s theorem and Lebesgue’s theorem,lim u → σ ( CX u , Y,
0) = σ ( X, Y, . Moreover using Proposition 3.6 in [14], for i ∈ { , . . . , n − } , we haveΛ i ( CX u , CX u ∩ B n ) b i = − g n − i − n Z G n − i − n χ ( CX u ∩ S n − ∩ H ) dH + 12 g n − i +1 n Z G n − i +1 n χ ( CX u ∩ S n − ∩ H ) dH. But χ ( CX u ∩ S n − ∩ H ) = χ ( CX u ∩ S n − u ∩ H ) = χ ( X ∩ S n − u ∩ H ) , for H ∈ G n − i +1 n or H ∈ G n − i − n , and lim u → χ ( CX u ∩ S n − ∩ H ) = χ (Lk( X ∩ H )).Passing to the limit as u → u → Λ i ( CX u , CX u ∩ B n ) b i = lim ǫ → Λ i ( X, X ∩ B nǫ ) b i ǫ i . The same proof works for i = n − i = n . Combining all these equalities, weget the result. (cid:3) Second step: X ⊂ R n is a closed conic definable set, ( Y, ⊂ ( R n ,
0) is a germof closed definable set.
Corollary 8.3.
Let X ⊂ R n be a closed conic definable set. For any germ of closeddefinable set ( Y, ⊂ ( R n , , the following principal kinematic formula holds: σ ( X, Y,
0) = n X i =0 Λ lim i ( X, · σ n − i ( Y, . Proof.
By Proposition 4.20, we know that σ ( X, Y,
0) = σ ( Y, X, P ni =0 Λ lim i ( X, · σ n − i ( Y,
0) = P ni =0 Λ lim i ( Y, · σ n − i ( X, (cid:3) Third step: ( X, ⊂ ( R n ,
0) and ( Y, ⊂ ( R n ,
0) are germs of closed definablesets.We assume that X and Y are included in an open neighborhood U of 0. Let ǫ > B nǫ ⊂ U . We setΓ X = (cid:26) ( 1 u x, u ) | x ∈ X, u ∈ ]0 , ǫ [ (cid:27) ⊂ R n +1 and Γ Y = (cid:26) ( 1 u y, u ) | y ∈ Y, u ∈ ]0 , ǫ [ (cid:27) ⊂ R n +1 . We recall that the tangent cones of X and Y are C X = Γ X ∩ R n × { } and C Y = Γ Y ∩ R n × { } . We will now define two tangent cones associated with b X and c Y v and will relate them to C X and C Y . Let c Γ X = (cid:26) ( 1 u x, t, u ) | x ∈ X, u ∈ ]0 , ǫ [ (cid:27) ⊂ R n +2 . The following lemma is easy to prove.
Lemma 8.4.
A point ( x, t ) belongs to c Γ X ∩ ( R n +1 × { } ) if and only if there is asequence of points ( x n , t n ) n ∈ N in b X and a sequence of positive real numbers ( u n ) n ∈ N such that u n → and ( x n u n , t n ) → ( x, t ) . Corollary 8.5.
We have [ C X = c Γ X ∩ ( R n +1 × { } ) .Proof. If ( x, t ) ∈ [ C X then there is a sequence of points ( x n ) n ∈ N in X and asequence of positive real numbers ( u n ) n ∈ N such that u n → x n u n → x . Applyingthe previous lemma to the sequences ( x n , t ) and ( u n ), we see that ( x, t ) ∈ c Γ X ∩ ( R n +1 × { } ).Conversely if ( x, t ) ∈ c Γ X ∩ ( R n +1 × { } ), 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 → x n u n , t n ) → ( x, t ). This implies that x ∈ C X and so that ( x, t ) ∈ [ C X . (cid:3) Let v ∈ S n − and let \ (Γ Y ) v = (cid:26) ( 1 u y, u t, u ) | ( y, t ) ∈ c Y v , u ∈ ]0 , ǫ [ (cid:27) ⊂ R n +2 . Lemma 8.6.
A point ( y, t ) belongs to \ (Γ Y ) v ∩ ( R n +1 × { } ) if and only if thereis a sequence of points ( y n , t n ) n ∈ N in c Y v and a sequence of positive real numbers ( u n ) n ∈ N such that u n → and ( y n u n , t n u n ) → ( y, t ) . Corollary 8.7.
We have \ ( C Y ) v = \ (Γ Y ) v ∩ ( R n +1 × { } ) .Proof. If ( y, t ) ∈ \ ( C Y ) v then there is a sequence of points ( y n ) n ∈ N in Y and asequence of positive real numbers ( u n ) n ∈ N such that u n → y n u n → y − tv . For n ∈ N , ( y n + u n tv, u n tv ) is in c Y v and ( y n + u n tvu n , u n tu n ) tends to ( y, t ). Therefore ( y, t )is in \ (Γ Y ) v ∩ ( R n +1 × { } ).Conversely if ( y, t ) is in \ (Γ Y ) v ∩ ( R n +1 × { } ), then there is a sequence of points( y n , t n ) n ∈ N in c Y v and a sequence of positive real numbers ( u n ) n ∈ N such that u n → y n u n , t n u n ) → ( y, t ). Then y n − t n v ∈ Y and y n − t n vu n tends to y − tv . So y − tv belongs to C Y . (cid:3) We note that C X = [ C X ∩ ( R n × { } ), C Y = \ ( C Y ) v ∩ ( R n × { } ) and that [ C X and \ ( C Y ) v are closed conic definable sets.Let us assume that X is equipped with a Whitney stratification S = { S i } li =0 with S = { } and 0 ∈ S i for i = 1 , . . . , l . We setΓ S i = n ( xu , u ) | x ∈ S i , u ∈ ]0 , ǫ [ o ⊂ R n +1 for i = 0 , . . . , s . Lemma 8.8.
The partition Γ X = ∪ si =0 Γ S i is a Whitney stratification of Γ X . RINCIPAL KINEMATIC FORMULAS FOR GERMS OF CLOSED DEFINABLE SETS 33
Proof.
The partition ∪ si =0 S i × ]0 , ǫ [ gives a Whitney stratification of X × ]0 , ǫ [. Let φ be the diffeomorphism φ ( x, u ) = ( u x, u ). We have Γ S i = φ ( S i × ]0 , ǫ [) for i =0 , . . . , s and Γ X = φ ( X × ]0 , ǫ [). This gives the result for Whitney’s conditions are C -invariant. (cid:3) We can equip C X with a definable stratification (Σ k ) l ′ k =0 where Σ = { } andΣ k is conic. This is possible for example by considering a Whitney stratification of C X ∩ S n − and extending it to C 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 X such that Σ ⊂ Γ S \ Γ S . The set of points x in Σ such that the Thom ( a u ) -condition is notsatisfied at x for the pair (Γ S , Σ) is a conic definable set of positive codimension.Proof. By [1] and [27], we already know that this set is definable of positive codi-mension 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 p n (Γ S ∩ { u = u n } ). Let λ > T λx Σ = T x Σ. Moreover q n = ( λy n , u n λ ) ∈ Γ S and T p n (Γ S ∩ { u = u n } ) = T q n (Γ S ∩ { u = u n λ } ). (cid:3) 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 [30] for the argument). This induces a Whitney stratification of c Γ X compatible with [ C X × { } . Namely if S ′ ⊂ R n +1 is a stratum Γ X then b S ′ = { ( x, t, u ) | ( x, u ) ∈ S ′ } is a stratum of c Γ X . Lemma 8.10.
This induced stratification of c Γ X satisfies the Thom ( a u ) -condition.Proof. Let ( x n , t n , u n ) n ∈ N be a sequence of points in c Γ X that tends to ( x, t, x n , t n , u n ) lies in a stratum c S ′ = { ( x ′ , t ′ , u ′ ) | ( x ′ , u ′ ) ∈ S ′ } andthat ( x, t, ∈ c S ′ , where c S ′ = { ( x ′ , t ′ , | ( x ′ , ∈ S ′ } . Since the pair ( S ′ , S ′ )satisfies the Thom ( a u )-condition, T ( x, S ′ ⊂ lim n → + ∞ T ( x n ,u n ) ( S ′ ∩ { u = u n } ).But T ( x n ,t n ,u n ) c S ′ = (cid:8) ( ν, τ, ξ ) | ( ν, ξ ) ∈ T ( x n ,u n ) S ′ (cid:9) , and T ( x,t, c S ′ = (cid:8) ( ν, τ, | ( ν, ∈ T ( x, S ′ (cid:9) . It is straightforward to conclude using the fact that T ( x n ,u n ) ( S ′ ∩ { u = u n } ) =( T ( x n ,u n ) S ′ ) ∩ { u = 0 } and T ( x n ,t n ,u n ) ( c S ′ ∩ { u = u n } ) = ( T ( x n ,t n ,u n ) c S ′ ) ∩ { u = 0 } if n is sufficiently big. (cid:3) Similarly we can equip Γ Y with a Whitney definable stratification compatiblewith C Y × { } , that satisfies the Thom ( a u )-condition and such that the strataof C Y are conic. This induces a Whitney stratification of \ (Γ Y ) v compatible with \ ( C Y ) v × { } . Namely if T ′ ⊂ R n +1 is a stratum of Γ Y then c 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. Proof.
The proof is the same as in the previous lemma, taking into account thefollowing remark: if c T ′ v = { ( y, t, u ) | ( y − tv, u ) ∈ T ′ } is a stratum of \ (Γ y ) v then T ( y,t,u ) c T ′ v = { ( ν, τ, ξ ) | ( ν − τ v, ξ ) ∈ T ( y − tv,u ) T ′ } . (cid:3) For 0 < u ≤ ǫ , we set X u = X ∩ S n − 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 X and C Y satisfy Condition (1). Thereexists a definable subset ∆ X,Y ⊂ S n − of positive codimension such that for v / ∈ ∆ X,Y , lim u → χ (cid:16) Lk( [ CX u ∩ c Y v ∩ { t ≥ } ) (cid:17) = χ (cid:16) Lk( b X ∩ c Y v ∩ { t ≥ } ) (cid:17) . Proof.
As in Lemma 8.1, we have to prove that χ (cid:16) Lk( [ CX u ∩ c Y v ∩ { t ≥ } ) (cid:17) = χ (cid:16) [ CX u ∩ c Y v ∩ { t ≥ } ∩ { ω = u } (cid:17) , if u is small enough, taking into account that c Y v is not conic.First let us fix v / ∈ ( − C Y ) ∩ S n − . There exist ǫ v > a > c Y v ∩ { t ≥ } ∩ B n +1 ǫ v ⊂ { ( x, t ) | ω ( x ) ≥ at } ∩ B n +1 ǫ v , which implies that there exists u > c Y v ∩ { t ≥ } ∩ { ω ≤ u } ∩ B n +1 ǫ v ⊂ ˚ B n +1 ǫ v . We have assumed that C X and C Y satisfy Condition (1). This means thattwo strata W and W ′ of C X and C Y (different from { } ) intersect transversally.Since these strata are conic, W ∩ { ω = 1 } and W ′ intersect transversally as welland so C X ∩ { ω = 1 } and C Y intersect transversally (in the stratified sense).As in Lemma 4.11, there exists a definable subset Γ C X ∩{ ω =1 } ,C Y ⊂ S n − of pos-itive codimension such that for v / ∈ Γ C X ∩{ ω =1 } ,C Y , \ C X ∩ { ω = 1 } and \ ( C Y ) v intersect transversally (in the stratified sense).We need a first auxiliary lemma. Lemma 8.13. If v / ∈ Γ C X ∩{ ω =1 } ,C Y , then there exists < u ≤ u such thatfor < u ≤ u and for ( x, t ) ∈ [ CX u ∩ c Y v ∩ { t > } ∩ { < ω ≤ u } ∩ B n +1 ǫ v , thesets [ CX u ∩ { t > } ∩ { ω = ω ( x ) } and c Y v ∩ { t > } intersect transversally (in thestratified sense) at ( x, t ) .Proof. Let us specify the stratifications we are working with. The set Y is equippedwith a Whitney stratification { T j } mj =0 , which induces a stratification { [ ( T j ) v } mj =0 of c Y v . Hence c Y v ∩ { t > } is stratified by { [ ( T j ) v ∩ { t > }} mj =0 . The set X is equippedwith a Whitney stratification { S i } li =0 . Hence for u small, CX u is stratified by { } ∪ { C ( S i ∩ S n − u ) } li =1 . As above this induces a stratification of [ CX u ∩ { t > } .We note that by the conic structure, the intersection [ CX u ∩ { t > } ∩ { ω = ω ( x ) } is always transverse (in the stratified sense) and the stratification of [ CX u ∩ { t > } ∩ { ω = ω ( x ) } is clear. RINCIPAL KINEMATIC FORMULAS FOR GERMS OF CLOSED DEFINABLE SETS 35
Assume that the above result is not true. Then we can find a sequence ofpositive real numbers ( u n ) n ∈ N that tends to 0 and a sequence of points ( x n , t n ) n ∈ N in \ CX u n ∩ c Y v ∩ { t > } ∩ { < ω ≤ u n } ∩ B n +1 ǫ such that \ CX u n ∩ { ω = ω ( x n ) } and c 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 c T v , where T is a stratum of Y . Moreover we can assume that there is a stratum S = { } of X such that for each n ∈ N , ( x n , t n ) ∈ \ CS u n . Since t n ω ( x n ) ≤ a , takinga subsequence if necessary, we can assume that ( x n ω ( x n ) , t n ω ( x n ) ) tends to ( x, t ). Let w n = u n x n ω ( x n ) , then w n ∈ CS u n ∩ { ω = u n } and so w n ∈ S ⊂ X . Therefore( w n , t n ω ( x n ) ) belongs to b X and ( u n w n , t n ω ( x n ) ) tends to ( x, t ). By Lemma 8.4 andCorollary 8.5, this implies that ( x, t ) is in [ C X . Moreover since ω ( w n u n ) = 1, ( x, t )belongs to [ C X ∩ { ω = 1 } . On the other hand, ( x n , t n ) ∈ c Y v and so by Lemma 8.6and Corollary 8.7, ( x, t ) belongs to \ ( C Y ) v .The points p n := ( u n w n , t n ω ( x n ) , u n ) belong to the stratum c Γ S of c Γ X and thepoint ( x, t ) belongs to a stratum b Σ of [ C X . By the Thom ( a u )-condition, we have T ( x,t ) b Σ ⊂ lim n → + ∞ T p n ( c Γ S ∩ { u = u n } ).Since b Σ is Σ × R and Σ is conic, b Σ intersects { ω = 1 } transversally. By the Thom( a u )-condition, { ω = 1 } intersects c Γ S ∩ { u = u n } transversally for n big enough,and so T ( x,t ) ( b Σ ∩ { ω = 1 } ) ⊂ lim n → + ∞ T p n ( c Γ S ∩ { ω = 1 } ∩ { u = u n } ) . But T p n ( c Γ S ∩ { ω = 1 } ∩ { u = u n } ) = T ( w n , tnω ( xn ) ) ( b S ∩ { ω = u n } ) and so T ( x,t ) ( b Σ ∩ { ω = 1 } ) ⊂ lim n → + ∞ T ( w n , tnω ( xn ) ) ( b S ∩ { ω = u n } ) . We note that T ( w n , tnω ( xn ) ) ( b S ∩ { ω = u n } ) = T ( x n ,t n ) ( \ CS u n ∩ { ω = ω ( x n ) } ) , by the conic structure of CS u n .The points q n := ( x n ω ( x n ) , t n ω ( x n ) , ω ( x n )) are in the stratum \ (Γ T ) v of \ (Γ Y ) v andthe point ( x, t ) is in a stratum c Σ ′ v of \ ( C Y ) v . By the Thom ( a u )-condition, we have T ( x,t ) c Σ ′ v ⊂ lim n → + ∞ T q n ( \ (Γ T ) v ∩ { u = ω ( x n ) } ). But T q n ( \ (Γ T ) v ∩ { u = ω ( x n ) } ) = T ( x n ,t n ) c T v and so T ( x,t ) c Σ ′ v ⊂ lim n → + ∞ T ( x n ,t n ) c T v .Since v / ∈ Γ C X ∩{ ω =1 } ,C Y , \ C X ∩ { ω = 1 } and \ ( C Y ) v intersect transversally(in the stratified sense). But \ C X ∩ { ω = 1 } = [ C X ∩ { ω = 1 } , and we concludethat T ( x,t ) ( b Σ ∩ { ω = 1 } ) + T ( x,t ) c Σ ′ v = R n +1 . Therefore lim n → + ∞ T ( x n ,t n ) ( \ CS u n ∩ { ω = ω ( x n ) } ) + lim n → + ∞ T ( x n ,t n ) c T v = R n +1 , and so, for n big enough T ( x n ,t n ) ( \ CS u n ∩ { ω = ω ( x n ) } ) + T ( x n ,t n ) c T v = R n +1 . This contradicts the construction of the sequence ( x n , t n ) and ends the proof of thisauxiliary lemma. (cid:3) Similarly the following second auxiliary lemma holds.
Lemma 8.14.
There exists < u ≤ u such that for < u ≤ u and for x ∈ CX u ∩ Y ∩ { < ω ≤ u } ∩ B nǫ v , the sets CX u ∩ { ω = ω ( x ) } and Y intersecttransversally (in the stratified sense) in R n at x . Let us choose u > u ≤ min { ǫ , u , u } , where ǫ is such that for0 < u ≤ ǫ , χ (cid:16) Lk( b X ∩ c Y v ∩ { t ≥ } ) (cid:17) = χ (cid:16) b X ∩ c Y v ∩ { t ≥ } ∩ { ω = u } (cid:17) . Then for ( x, t ) ∈ [ CX u ∩ c Y v ∩{ < ω ≤ u }∩{ t > }∩ B n +1 ǫ v , [ CX u ∩{ ω = ω ( x ) }∩{ t > } and c Y v ∩ { t > } intersect transversally (in the stratified sense) at ( x, t ). Thisimplies that [ CX u ∩{ t > } and c Y v ∩{ t > } intersect transversally at ( x, t ) and that { ω = ω ( x ) } intersects [ CX u ∩ c Y v ∩ { t > } transversally at ( x, t ), and so ( x, t ) is nota stratified critical point of ω | [ CX u ∩ c Y v ∩{ t> } . Similarly if ( x, ∈ [ CX u ∩ c Y v ∩ { <ω ≤ u }∩ B n +1 ǫ v , then ( x,
0) is not a stratified critical point of ω | [ CX u ∩ c Y v ∩{ t =0 } . Hencewe conclude that ω : [ CX u ∩ c Y v ∩ { < ω ≤ u } ∩ { t ≥ } ∩ B n +1 ǫ v → R is a stratifiedsubmersion and so that χ (cid:16) Lk( [ CX u ∩ c Y v ∩ { t ≥ } ) (cid:17) = χ (cid:16) [ CX u ∩ c Y v ∩ { t ≥ } ∩ { ω = u } (cid:17) . Therefore if dim C Y ≤ n −
1, we can take∆
X,Y = (cid:0) ( − C Y ) ∩ S n − (cid:1) ∪ Γ C X ∩{ ω =1 } ,C Y . If dim C Y = n then dim Y = n . Let Y ′ be the union of the strata of Y ofdimension less than or equal to n −
1. If v ∈ ( − C Y ) \ ( − C Y ′ ), we know that χ (cid:16) Lk( [ CX u ∩ c Y v ∩ { t ≥ } ) (cid:17) = 1 + χ c (cid:18) [ CX u ∩ c Y v ∩ { t ≥ } ∩ { ω = ǫ, t < a ǫ } (cid:19) ,χ (cid:16) Lk( b X ∩ c Y v ∩ { t ≥ } ) (cid:17) = 1 + χ c (cid:18) b X ∩ c Y v ∩ { t ≥ } ∩ { ω = ǫ, t < a ǫ } (cid:19) , for 0 < ǫ ≪ a > c Y ′ v ∩ { t ≥ } ⊂ { ( x, t ) | ω ( x ) ≥ at } in aneighborhood of (0 , ǫ > < u ≤ ǫ , χ (cid:16) Lk( b X ∩ c Y v ∩ { t ≥ } ) (cid:17) = 1 + χ c (cid:18) b X ∩ c Y v ∩ { t ≥ } ∩ { ω = u, t < a u } (cid:19) . Since χ c (cid:18) b X ∩ c Y v ∩ { t ≥ } ∩ { ω = u, t < a u } (cid:19) = χ c (cid:18) [ CX u ∩ c Y v ∩ { t ≥ } ∩ { ω = u, t < a u } (cid:19) , we have to prove that χ (cid:16) Lk( [ CX u ∩ c Y v ∩ { t ≥ } ) (cid:17) = 1 + χ c (cid:18) [ CX u ∩ c Y v ∩ { t ≥ } ∩ { ω = u, t < a u } (cid:19) , RINCIPAL KINEMATIC FORMULAS FOR GERMS OF CLOSED DEFINABLE SETS 37 if u is small enough. By the previous case, we know that for u small enough andfor v / ∈ Γ C X ∩{ ω =1 } ,C Y ′ , ω : [ CX u ∩ c Y ′ v ∩ { < ω ≤ u } ∩ { t ≥ } ∩ B n +1 ǫ → R is a stratified submersion, for an appropriate ǫ >
0. Since the strata of c Y v \ c Y ′ v havedimension n + 1 and the strata of Y \ Y ′ have dimension n , ω : [ CX u ∩ c Y v ∩ { < ω ≤ u } ∩ { t ≥ } ∩ B n +1 ǫ → R is a stratified submersion by the conic structure of CX u . For the same reason andbecause c Y ′ v ∩ { t ≥ } ∩ { t = a ω ( x ) } = { (0 , } , we see that ω : [ CX u ∩ c Y v ∩ { < ω ≤ u } ∩ { t = 2 a ω ( x ) } → R is a stratified submersion. Hence for 0 < ǫ ≤ u , χ c (cid:18) [ CX u ∩ c Y v ∩ { t ≥ } ∩ { ω = u, t < a u } (cid:19) = χ c (cid:18) [ CX u ∩ c Y v ∩ { t ≥ } ∩ { ω = ǫ, t < a ǫ } (cid:19) . If dim C Y = n , we take∆ X,Y = (cid:0) ( − C Y ′ ) ∩ S n − (cid:1) ∪ Γ C X ∩{ ω =1 } ,C Y ′ . (cid:3) Theorem 8.15.
Let ( X, ⊂ ( R n , and ( Y, ⊂ ( R n , be two germs of closeddefinable sets. The following principal kinematic formula holds: σ ( X, Y,
0) = n X i =0 Λ lim i ( X, · σ n − i ( Y, . Proof.
By Lemma 4.10, there exists a definable subset Σ
X,Y ⊂ SO ( n ) of positivecodimension such that for γ / ∈ Σ X,Y , X and γY satisfy Condition (1). Let usfix γ / ∈ Σ X,Y . By Lemma 4.11, there exists a definable subset Γ
X,γY ⊂ S n − ofpositive codimension such that for v / ∈ Γ X,γY , v satisfies Condition (2). Let uschoose v / ∈ Γ X,γY . By Lemma 4.2, the function t : b X ∩ \ ( γY ) v → R has an isolatedstratified critical point at (0 , ǫ → lim δ → + χ ( X ∩ ( γY + δv ) ∩ B nǫ ) = χ (cid:16) Lk( b X ∩ \ ( γY ) v ∩ { t ≥ } ) (cid:17) , and so σ ( X, Y,
0) = 1 s n − Z SO ( n ) × S n − χ (cid:16) Lk( b X ∩ \ ( γY ) v ∩ { t ≥ } ) (cid:17) dvdγ. Of course the same equality is true if we replace X with CX u . By Corollary 8.3,for 0 < u ≤ ǫ , we have σ ( CX u , Y,
0) = n X i =0 Λ i ( CX u , CX u ∩ B n ) b i · σ n − i ( Y, . Since C ( γY ) = γ ( C Y ), by Lemma 4.10 there exists a definable subset Σ C X,C Y ⊂ SO ( n ) of positive codimension such that for γ / ∈ Σ C X,C Y , C X and C ( γY ) satisfyCondition (1). For γ / ∈ Σ C X,C Y and v / ∈ ∆ X,γY ,lim u → χ (cid:16) Lk( [ CX u ∩ \ ( γY ) v ∩ { t ≥ } ) (cid:17) = χ (cid:16) Lk( b X ∩ \ ( γY ) v ∩ { t ≥ } ) (cid:17) . Hence, by Hardt’s theorem and Lebesgue’s theorem,lim u → σ ( CX u , Y,
0) = σ ( X, Y, . We end the proof as in Proposition 8.2. (cid:3)
Let us specify this kinematic formula when d + e = n , d = dim X and e = dim Y .We denote by X d (resp. Y e ) the union of the top-dimensional strata of X (resp. Y ). Corollary 8.16.
Let ( X, ⊂ ( R n , and ( Y, ⊂ ( R n , be two germs of closeddefinable sets such that d + e = n , where d = dim X and e = dim Y .The followingformula holds: s n − Z SO ( n ) × S n − lim ǫ → lim δ → + (cid:0) X d ∩ ( γY e + δv ) ∩ B nǫ (cid:1) dγdv = Θ d ( X ) · Θ e ( Y ) . Proof.
For γ generic in SO ( n ) and v generic in S n − ,lim ǫ → lim δ → + χ ( X ∩ ( γY + δv ) ∩ B nǫ ) = lim ǫ → lim δ → + (cid:0) X d ∩ ( γY e + δv ) ∩ B nǫ (cid:1) . (cid:3) Let us formulate now the second principal kinematic formula.
Theorem 8.17.
Let ( X, ⊂ ( R n , and ( Y, ⊂ ( R n , be two germs of closeddefinable sets. The following principal kinematic formula holds: Λ lim0 ( X, Y,
0) = n X i =0 Λ lim i ( X, · Λ lim n − i ( Y, . Proof.
Let us compute the integrals1 s n − Z SO ( n ) χ (Lk( X ∩ γY )) dγ, and 1 s n − Z SO ( n ) Z S n − χ (Lk( X ∩ γY ∩ { u ∗ = 0 } )) dudγ. Let us assume first that X and Y are conic closed definable sets. We have alreadycomputed the first integral in the proof of Proposition 5.1 and we have found that1 s n − Z SO ( n ) χ (Lk( X ∩ γY )) dγ = n − X i =0 ˜Λ i (Lk( X ) , Lk( X )) s i · g i +1 n Z G i +1 n χ (Lk( Y ∩ H )) dH, which can be rewritten in the following way:1 s n − Z SO ( n ) χ (Lk( X ∩ γY )) dγ RINCIPAL KINEMATIC FORMULAS FOR GERMS OF CLOSED DEFINABLE SETS 39 = n X i =1 Λ i ( X, X ∩ B n ) b i · g in Z G in χ (Lk( Y ∩ H )) dH. The same computation applied to X ∩ { u ∗ = 0 } yields1 s n − Z SO ( n ) Z S n − χ (Lk( X ∩ γY ∩ { u ∗ = 0 } )) dudγ = n − X i =0 (cid:16) s n − Z S n − ˜Λ i (Lk( X ∩ { u ∗ = 0 } ) , Lk( X ∩ { u ∗ = 0 } )) s i du × g i +1 n Z G i +1 n χ (Lk( Y ∩ H )) dH (cid:17) = n − X i =0 (cid:16) g n − n Z G n − n ˜Λ i (Lk( X ∩ L ) , Lk( X ∩ L )) s i dL × g i +1 n Z G i +1 n χ (Lk( Y ∩ H )) dH. (cid:17) Using the notations and normalizations of [3], Theorem 4.4, we can write1 g n − n Z G n − n ˜Λ i (Lk( X ∩ L ) , Lk( X ∩ L )) s i dL = 1 s n − Z SO ( n ) ˜Λ i (Lk( X ∩ γE ) , Lk( X ∩ γE )) s i dγ, where E is a ( n − S n − . By the spherical kinematicformula, we find that1 s n − Z SO ( n ) ˜Λ i (Lk ( X ∩ γE ) , Lk( X ∩ γE )) s i dγ = 1 s i +1 ˜Λ i +1 (Lk( X ) , Lk( X ))= 1 b i +2 Λ i +2 ( X, X ∩ B n ) . Hence we get that1 s n − Z SO ( n ) Z S n − χ (Lk( X ∩ γY ∩ { u ∗ = 0 } )) dvdγ = n X i =2 ˜Λ i ( X, X ∩ B n ) b i · g i − n Z G i − n χ (Lk( Y ∩ H )) dH. Then we apply this result to CX u and CY u where X u = X ∩ S n − u and Y u = Y ∩ S n − u ,and make u → s n − Z SO ( n ) χ (Lk( X ∩ γY )) dγ = n X i =1 lim ǫ → Λ i ( X, X ∩ B nǫ ) b i ǫ i · g in Z G in χ (Lk( Y ∩ H )) dH, and 1 s n − Z SO ( n ) Z S n − χ (Lk( X ∩ γY ∩ { u ∗ = 0 } )) dudγ = n X i =2 lim ǫ → Λ i ( X, X ∩ B nǫ ) b i ǫ i · g i − n Z G i − n χ (Lk( Y ∩ H )) dH. Therefore by the relation between σ ( X, Y,
0) and Λ ( X, Y,
0) and by Theorem 8.15,we getΛ ( X, Y,
0) = Λ lim0 ( X, σ n ( Y, lim1 ( X, σ n − ( Y, − g n Z G n χ (Lk( Y ∩ H )) dH ! + n X i =2 Λ lim i ( X, · A i , where A i = σ n − i ( Y, − g in Z G in χ (Lk( Y ∩ H )) dH − g i − n Z G i − n χ (Lk( Y ∩ H )) dH. By [15], Theorem 5.6 and its proof, we have that σ n ( Y,
0) = Λ lim n ( Y,
0) and for i ≥ σ n − i ( Y,
0) = 12 g i +1 n Z G i +1 n χ (Lk( Y ∩ H )) dH + 12 g in Z G in χ (Lk( Y ∩ H )) dH. Moreover by [14], Theorem 5.1, we have thatΛ lim n − ( Y,
0) = 12 g n Z G n χ (Lk( Y ∩ H )) dH and for i ≥ lim n − i ( Y,
0) = 12 g i +1 n Z G i +1 n χ (Lk( Y ∩ H )) dH − g i − n Z G i − n χ (Lk( Y ∩ H )) dH. These equalities enable us to end the proof. (cid:3)
For Y = H , where H ∈ G n − kn and k ∈ { , . . . , n } , the above kinematic formulawrites σ ( X, H,
0) = Λ lim k ( X, . Hence we recover our Theorem 3.7, because for H ∈ G n − kn , β ( H ) = 1 s n − Z S n − lim ǫ → lim δ → + Λ ( H δ,v ∩ X, H δ,v ∩ X ∩ B nǫ ) dv, by the co-area formula. 9. More kinematic formulas
In view of Theorem 8.17, a natural question is to express the following sums X i + j = p + n Λ i ( X, · Λ j ( X, , for k = 1 , . . . , n as the right-hand side of a kinematic formula. The answer is quitesimple and explained briefly in the next proposition. Proposition 9.1.
Let ( X, ⊂ ( R n , and ( Y, ⊂ ( R n , be two germs of closeddefinable sets. For k ∈ { , . . . , n } , the following kinematic formula holds: Z SO ( n ) lim ǫ → Λ k ( X ∩ γY, X ∩ γY ∩ B nǫ ) b k ǫ k dγ = X i + j = k + n Λ lim i ( X, · Λ lim j ( Y, . RINCIPAL KINEMATIC FORMULAS FOR GERMS OF CLOSED DEFINABLE SETS 41
Proof.
When X and Y are conic, it enough to apply the spherical kinematic formu-las. The general case can be deduced as we have already done in several previousproofs. (cid:3) Corollary 9.2.
Let ( X, ⊂ ( R n , and ( Y, ⊂ ( R n , be two germs of closeddefinable sets. The following principal kinematic formula holds: Z SO ( n ) lim ǫ → Λ ( X ∩ γY, X ∩ γY ∩ B nǫ ) dγ = n X i =0 Λ lim i ( X, · n − i X j =0 Λ lim j ( Y, . Proof.
Apply Proposition 9.1 and Corollary 3.6. (cid:3)
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