Intersections and translative integral formulas for boundaries of convex bodies

Transcription

1 to appear in Math. Nachr. (199?), Intersections translative integral formulas for boundaries of convex bodies By Daniel Hug Reiner Schätzle of Freiburg (Received November 06, 1998) Abstract. Let K, L IR n be two convex bodies with non-empty interiors with boundaries K, L, let χ denote the Euler characteristic as defined in singular homology theory. We prove two translative integral formulas involving boundaries of convex bodies. It is shown that the integrals of the functions t χ( K ( L + t)) t χ( K (L + t)), t IR n, with respect to an n- dimensional Haar measure of IR n can be expressed in terms of certain mixed volumes of K L. In the particular case where K L are outer parallel bodies of convex bodies at distance r > 0, the result will be deduced from a recent (local) translative integral formula for sets with positive reach. The general case follows from this from the following (global) topological result. Let K r, L r denote the outer parallel bodies of K, L at distance r 0. Establishing a conjecture of Firey (1978), we show that the homotopy type of K r L r K r L r, respectively, is independent of r 0 if K L if K L intersect almost transversally. As an immediate consequence of our translative integral formulas, we obtain a proof for two kinematic formulas which have also been conjectured by Firey. 1. Introduction Kinematic formulas play a central rôle in integral stochastic geometry. In general, a kinematic formula can be described in the following abstract setting. Let X be a topological space, let G be a topological group, let ϕ : G X X be a transitive, continuous left action of G on X such that ϕ(, x) : G X is an open map for any x X. In this situation, (X, ϕ) is called a homogeneous G-space; confer [28], p. 15. Usually, it is assumed in addition that G X are locally compact Hausdorff (see [9], p. 121). Under this additional assumption on G, there always exists a left Haar measure µ on G. Further, let γ be a geometric functional on certain 1991 Mathematics Subject Classification. Primary 52A20, 52A22; Secondary 49J52, 53C65, 55P10. Keywords phrases. Translative integral formula, integral geometry, convex surface, transversal intersection, homotopy type, Euler characteristic, Lipschitz manifold.

2 2 Math. Nachr. (199?) subsets of X, let A, B be suitable subsets of X such that γ(a gb) is defined for µ-almost all g G g γ(a gb) is integrable with respect to µ. In this general setting, a kinematic formula allows one to express the integral (1.1) γ(a gb)µ(dg) G in terms of geometric functionals of A B. Such kinematic formulas, for various choices of G, X γ for different classes of subsets of X, have been the subject of numerous investigations. Moreover, operations different from the formation of the intersection of A gb or functions γ which are measure-valued have been considered in integral geometry. We refer to the surveys in [25, 4.5], [29], [28, 3], to [31, V], [21], [10], [20], [17] for a detailed discussion of the extensive literature on this subject. The basic result in this context is the principal kinematic formula, which is due to Blaschke Santaló. A differential-geometric version for domains in IR n which are bounded by imbedded hypersurfaces of class C 2 was proved by Chern Yien. In the setting of convex geometry, the principal kinematic formula corresponds to the case where X is the Euclidean space IR n with n 2, scalar product, induced norm, G is the group G(n) of proper rigid motions of IR n, ϕ(g, x) = g(x) for g G(n) x IR n, γ is the Euler characteristic χ, A, B are convex bodies (compact convex sets with non-empty interiors) K, L IR n. Let α(i) denote the volume of an i-dimensional Euclidean unit ball, normalize µ as in [25]. To [25] we also refer for all notions of convex geometry which are not explicitly defined here. Then the integral (1.1) can be expressed in terms of the quermassintegrals (Minkowski functionals) W i (K), W j (L), i, j = 0,..., n, of K L (see [25], 4.2 (5.3.19)), namely (1.2) G(n) χ(k gl)µ(dg) = 1 α(n) n k=0 ( ) n W n k (K)W k (L). k In translative integral geometry, again X is the Euclidean space IR n but now G is the translation group of IR n (which will be identified with IR n ) ϕ(t, x) = x + t for t, x IR n. First contributions to translative integral geometry are due to Blaschke [1] Berwald Varga [2]. But only within the last 15 years translative integral formulas have been investigated systematically. Progress in this field was accompanied by, in fact required for, a corresponding development of stochastic geometry, where non-isotropic models naturally appear in applications. Main recent contributions are due to Schneider Weil [27], Weil [34], [35], Rataj M. Zähle [19], Rataj [18]. A particular feature of such translative formulas is that certain mixed functionals such as mixed volumes or mixed curvature measures of the sets involved cannot be avoided. The mixed volume of convex bodies K 1,..., K n in IR n is denoted by V (K 1,..., K n ). In particular, we write V (K[k], L[n k]) if K 1 =... = K k = K K k+1 =... = K n = L, where k {0,..., n} K, L are convex bodies. The defining relation for these special mixed volumes is H n (λk + µl) = n k=0 ( ) n λ k µ n k V (K[k], L[n k]), k

3 Hug/Schätzle, Intersections translative integral formulas 3 where H n denotes the n-dimensional Hausdorff measure λ, µ 0. In particular, for k = 0,..., n the Euclidean unit ball B one obtains W n k (K) = V (K[k], B[n k]). All basic properties of mixed volumes which we need subsequently are described in Section 5 of Schneider s book [25]. Now the translative counterpart to formula (1.2) is given by n ( ) n (1.3) χ(k (L + t))h n (dt) = V (K[k], L[n k]); IR n k see 5.3 in [25]. Extensions of formulas (1.2) (1.3) to finite unions of convex bodies follow immediately from well-known properties of valuations on convex bodies. More explicit information extensions to certain finite unions of sets with positive reach have been obtained by means of certain index functions which take account of multiplicities; see, e.g., Schneider [23] M. Zähle [37]. In general, however, boundaries of arbitrary convex bodies cannot be represented as locally finite unions of sets with positive reach. It is one purpose of the present paper to establish the following analogues of formula (1.3) involving boundaries of convex bodies. Moreover, we study related topological questions concerning intersections of boundaries of convex bodies. One difficulty in this context is that the Euler characteristic of the intersections K ( L+t) K (L+t) need not be defined for all translation vectors t IR n (see the discussion below). This indicates that for the proof of the following two theorems topological measure theoretic arguments have to be combined. Theorem 1.1. Let K, L IR n be convex bodies. Then the map t χ( K ( L+t)) is integrable with respect to H n on IR n (1.4) χ( K ( L + t))h n (dt) IR n n 1 ( ) n {V = (1 + ( 1) n ) (K[k], L[n k]) + ( 1) k 1 V (K[k], L[n k]) }. k k=0 k=1 Theorem 1.2. Let K, L IR n be convex bodies. Then the map t χ( K (L+t)) is integrable with respect to H n on IR n (1.5) χ( K (L + t))h n (dt) IR n n 1 ( ) n {V = (K[k], L[n k]) + ( 1) n 1 k V (K[k], L[n k]) }. k Kinematic formulas, concerning the group of proper rigid motions, can be deduced from the preceding two theorems by first applying the corresponding translative formulas to K ρl, where ρ is an arbitrary proper rotation, by subsequently carrying out the integration with respect to the Haar probability measure on the k=0

4 4 Math. Nachr. (199?) group of proper rotations SO(n) of IR n. The desired results then follow by an application of equation (5.3.25) in Schneider s book [25]. As an immediate consequence we thus obtain for convex bodies K, L IR n the following kinematic formulas which have been conjectured by Firey (1978): (1.6) G(n) if n is even, (1.7) if n is odd; moreover, (1.8) G(n) χ( K g L)µ(dg) = 2 α(n) G(n) χ( K gl)µ(dg) = 1 α(n) n 1 k=1 ( ) n (1 ( 1) k ) W n k(k)w k (L) k χ( K g L)µ(dg) = 0 n 1 k=0 ( ) n (1 ( 1) n k ) W n k(k)w k (L) k for all n 2. Alternatively, formulas (1.6) (1.8) can also be deduced from Theorem (see also Corollary 2.2.2) of Fu in [11], but the theory developed in [10], [11] does not cover the present more general results of translative integral geometry. The basic idea for the proof of Theorems goes back to Firey s suggestion for a proof of (1.6) (1.8). To describe the idea, first in this special case, let again B denote the Euclidean unit ball. For a convex body C IR n r 0, we write C r := C + rb call C r a parallel body of C at distance r. It is easy to see that for convex bodies D, E IR n, r > 0, K := D r L := E r the boundaries K L are sets of positive reach. It can be deduced from a special case of Federer s kinematic formula for sets with positive reach [8], Theorem 6.11, that the kinematic formulas (1.6) (1.8) hold for such a special choice of K L. Then one passes to the limit r 0 in these formulas. The right-h sides behave continuously with respect to this limit. However, for the integrals this is not at all obvious. The same problem arises in the proofs of the corresponding translative integral formulas. Moreover, in the general case, Federer s kinematic formula cannot be applied; instead we use a recent translative integral formula, due to Rataj M. Zähle [19]. As a first step towards a proof of Theorems , one has to ensure that χ( K ( L + t)) χ( K (L + t)) are well-defined, that is, the homology of the respective topological spaces is finitely generated, at least for H n -almost all t IR n. In Section 2, it will be shown that K ( L + t) is an (n 2)-dimensional compact Lipschitz manifold without boundary K (L + t) is an (n 1)-dimensional compact Lipschitz manifold with boundary, provided that the boundaries of K L + t intersect almost transversally (see below) K (L + t). Then it follows from Propositions IV.8.10 V.4.11 in [5] that the homology of these intersections is finitely generated; confer also Corollary VIII.1.4 in [5] for the case of a topological manifold without boundary. In order to give a precise definition for the notion of transversality employed in this paper, let us denote by N(K, x) the closed convex cone of exterior normal vectors of

5 Hug/Schätzle, Intersections translative integral formulas 5 a convex body K at the boundary point x K. For convex bodies K, L IR n, we say that K L intersect almost transversally if (1.9) N(K, x) N(L, x) = {0} (1.10) N(K, x) ( N(L, x)) = {0} hold for all x K L. A classical result of Federer [8] or, alternatively, a recent result of Schneider [26] then imply that K g L intersect almost transversally for µ-almost all g G(n). In fact, Schneider s result even yields that the boundaries intersect transversally for µ-almost all g G(n). Here we say that the boundaries of two convex bodies K L intersect transversally if for all common boundary points x K L the intersection of the linear hulls of the normal cones N(K, x) N(L, x) equals {0}, that is (1.11) lin N(K, x) lin N(L, x) = {0} for all x K L. Certainly, (1.11) is a much stronger condition than (1.9) (1.10), since we do not adopt any smoothness assumptions for K L. In order to prove translative integral formulas, however, we have to ensure that K L + t intersect almost transversally for H n -almost all t IR n. In fact, such a result is implicitly contained in a paper by Ewald, Larman Rogers [7]. Some assumptions of convexity seem to be essential in this context. We also wish to emphasize that the proof of an integral formula often requires some delicate argument which allows to exclude a set of measure zero; confer for instance Federer [8], Schneider [22], [24], Fu [11], Glasauer [12], [13], [14], Rataj Zähle [19], Zähle [38]. As a consequence of the preceding discussion we can conclude that the Euler characteristics of the intersections K ( L + t) K (L + t) are well-defined for H n -almost all t IR n. For a proof of formulas (1.6) (1.8), Firey further suggested to show that the Euler characteristics of the sets K r g L r K r gl r, respectively, are independent of r > 0, at least for µ-almost all g G(n). It turns out, however, that in order to follow this strategy of proof it is necessary to establish the corresponding statement for all r 0. Actually, the main problem is to include the case r = 0. More generally, the following two theorems will be proved. In the first of these we consider the intersections of the boundaries of the parallel bodies of K L. Theorem 1.3. Let K, L IR n, n 2, be two convex bodies with common interior whose boundaries intersect almost transversally, that is K L, (1.9) (1.10) are satisfied. Define the intersections S r := K r L r, r 0. Then all S r have the same homotopy type, that is S r S s for all r, s 0.

6 6 Math. Nachr. (199?) Moreover, the Euler characteristic χ(s r ) is well-defined for r 0 χ(s r ) = χ(s s ) for all r, s 0. For the intersections of boundaries bodies, we shall prove: Theorem 1.4. Let K, L IR n, n 2, be two convex bodies with K L for which (1.9) (1.10) are satisfied. Define the intersections H r := K r L r, r 0. Then all H r have the same homotopy type, that is H r H s for all r, s 0. Moreover, the Euler characteristic χ(h r ) is well-defined for r 0 χ(h r ) = χ(h s ) for all r, s 0. Theorems are proved in Sections These two theorems are established in three steps. In Section 2.1, we show with an implicit function theorem for Lipschitz functions that the union (1.12) M := 0 r 1 is an (n 1)-dimensional topological, actually a compact Lipschitz, manifold with boundary M = S 0 S 1. Outside K L one knows that the distance functions d(k, ) d(l, ) have continuous gradients, which yields that the intersections S r, for r > 0, are (n 2)-dimensional C 1 -manifolds whose normal space is spanned by d(k, ) d(l, ). In a Morse type lemma, using a flow induced by these normals, we conclude in Section 2.2 that the sets S r are homeomorphic for r > 0. This flow can be extended to r = 0, thus we get a surjective, continuous map S r ϕ : S 1 S 0. As the normals need not be unique for the given convex bodies, ϕ need not be injective induces, being a quotient map, a possibly non trivial equivalence relation on S 1. The quotient map ϕ enables us to show that S 0 is a strong deformation retract of M, hence (1.13) S 0 M. This can also be seen observing that M is the mapping cylinder of ϕ; see [30, 1.4]. In Section 2.3, we use a result of Brown on collarings for boundaries of topological manifolds (see [3]), which yields that the inclusion i : M M M

7 Hug/Schätzle, Intersections translative integral formulas 7 is a homotopy equivalence, that is, M M is a weak deformation retract of M. As clearly the Morse type lemma shows that S 1 is a strong deformation retract of 2 M M, we obtain S 1 M M M. 2 Together with (1.13), this will prove Theorem 1.3. The proof of Theorem 1.4 follows the same line. In Section 2.1, we prove that the union (1.14) N := 0 r 1 is an n-dimensional compact Lipschitz manifold with boundary H r N = M H 0 H 1. In Section 2.4, we modify the flow used in Section 2.2 in order to prove that the sets H r are homeomorphic for r > 0 that H 0 is a strong deformation retract of N, in particular (1.15) H 0 N. Finally, we apply once again Brown s Theorem to conclude the proof of Theorem Invariance of the homotopy type In this section, we consider two convex bodies K, L IR n, n 2, which have common interior points, that is (2.1) K L, which intersect almost transversally. Let d(c, ) : IR n [0, ) denote the distance function of a non-empty set C IR n. In particular, we consider the parallel bodies K r := {x IR n d(k, x) r}, L r := {x IR n d(l, x) r}, for r 0, which are again convex bodies which can also be described as K + rb, L + rb. Note that (2.1) implies that K r L s have common interior points, too, hence (2.2) N(K r, x) ( N(L s, x)) = {0} holds for all r, s 0 all x K r L s. In the following, we write A B, for A, B IR n, to denote the set of all points which are in A not in B. The Minkowski sum of A B (the reflection of B in the origin) is therefore denoted by A + ( B). Moreover, if f is any function which is defined on a subset Ω IR n takes its values in IR, then we write [f = r], r IR, for the set of all x Ω for which f(x) = r, etc. Subsequently, we shall use the notion of a Lipschitz manifold with or without a boundary. For the definition of a Lipschitz manifold (without boundary) we refer to

8 8 Math. Nachr. (199?) [32] or [33]. Actually, what we shall call a Lipschitz manifold will even be a locally Lipschitz graph in IR n, hence a strong Lipschitz submanifold of IR n in the sense of Walter. The notion of a Lipschitz manifold with boundary is then defined in the obvious way Lipschitz manifolds In this subsection, we assume that K, L are convex bodies which satisfy (2.1), (1.9) (1.10). The following lemma will be needed. Lemma 2.1. Assume that N(K, x) N(L, x) = {0} for all x K L. Then N(K r, x) N(L r, x) = {0}, for all r 0 x K r L r. Proof. Suppose there is some r > 0 some x K r L r such that u N(K r, x) N(L r, x) u = 1. Let B(x, r), for x IR n r 0, denote the closed ball of radius r centred at x. Then B(x, r) K = {x ru} = B(x, r) L. In fact, B(x, r) K follows from x K r. Moreover, (B(x, r) K) {x ru} implies that x K r or u / N(K r, x). Therefore, x ru K L u N(K, x ru) N(L, x ru), a contradiction. The main two results of this subsection are Propositions Proposition 2.2. Under the assumptions (2.1), (1.9) (1.10), the set M, defined in (1.12), is an (n 1)-dimensional compact Lipschitz manifold with boundary M = S 0 S 1, S r, for any r 0, is an (n 2)-dimensional compact Lipschitz manifold without boundary. Proof. Let z 0 M choose p 0 K L. By an appropriate choice of a coordinate system, we can ensure that p 0 z 0 (0, )e n, where (e 1,..., e n ) is an orthonormal basis of IR n. Set r 0 := d(k, z 0 ) = d(l, z 0 ) [0, 1]. Lemma 2.1 yields that (2.3) N(K r0, z 0 ) N(L r0, z 0 ) = {0}. Let δ > 0 be chosen such that B(p 0, 3δ) K L. Further, we write p 0 = (x 0, t 1 ) z 0 = (x 0, t 0 ), where x 0 IR n 1, t 0, t 1 IR t 0 < t 1. Then, for any r 0, there is a uniquely determined convex function f(, r) such that K r (B(p 0, 3δ) + (, 0)e n ) = graph f(, r) = {(x, f(x, r)) x B (x 0, 3δ)}, where B (x 0, 3δ) IR n 1 IR n 1 {0} denotes a closed ball of radius 3δ centred at x 0 in IR n 1. Similarly, L r is the graph of a convex function g(, r) on B (x 0, 3δ) for

9 Hug/Schätzle, Intersections translative integral formulas 9 all r 0. Obviously, we have f(x 0, r 0 ) = t 0 = g(x 0, r 0 ). Claim 1. Let c > 0. Then f C 0,1 (B (x 0, δ) [0, c]). Proof of Claim 1. Since K c L c are compact, there is a number a > 0 such that f(x, r) a holds for x B (x 0, 3δ) r [0, c]. Then, as in the proof of Theorem in [25], we obtain that (2.4) f(x 1, r) f(x 2, r) 2a δ x 2 x 1, for x 1, x 2 B (x 0, 2δ) r [0, c]. Fix some x B (x 0, δ) for the moment, choose r 1, r 2 [0, c] with 0 < r 2 r 1 < δ. For (x, f(x, r 2 )) K r2 there is some q K r1 such that (x, f(x, r 2 )) q = r 2 r 1. From r 2 r 1 < δ x B (x 0, δ) we infer that q can be represented as q = (x 3, f(x 3, r 1 )) with x 3 B (x 0, 2δ). Hence, we obtain f(x, r 2 ) f(x, r 1 ) f(x, r 2 ) f(x 3, r 1 ) + f(x 3, r 1 ) f(x, r 1 ) r 2 r 1 + 2a ( ) 2a δ x 3 x δ + 1 r 2 r 1. On the other h, if r 2 r 1 > δ, then Thus, (2.5) f(x, r 2 ) f(x, r 1 ) 2a 2a δ r 2 r 1. f(x, r 2 ) f(x, r 1 ) ( ) 2a δ + 1 r 2 r 1, for any x B (x 0, δ) r 1, r 2 [0, c]. Now (2.4), (2.5) the triangle inequality imply the assertion of Claim 1. We set ˆf(x) := f(x, r 0 ) ĝ(x) := g(x, r 0 ). Since ˆf ĝ are convex functions, the subdifferentials ˆf(x 0 ) ĝ(x 0 ) are defined as in convex geometry; confer [25, p. 30] or [4, p. 36 Proposition 2.2.7]. By (2.3) [4, p. 61, Corollary] (see also [25, Theorem ]), we obtain that ˆf(x 0 ) ĝ(x 0 ) =. Moreover, ˆf(x 0 ) ĝ(x 0 ) are non-empty compact convex subsets of IR n 1 ; confer [4, Proposition 2.1.2]. Hence, we can find u 0 IR n 1 γ 1, γ 2 IR such that (2.6) v, u 0 γ 1 > γ 2 w, u 0 for all v ˆf(x 0 ) w ĝ(x 0 ). Next we extend the definition of the functions f g by setting f(x, r) := f(x, r) g(x, r) := g(x, r) for x B (x 0, δ) r 0. Then f(, r) is convex for all r IR, in addition, f C 0,1 (B (x 0, δ) [ c, c]) for all c > 0. Then we define Γ := f g.

10 10 Math. Nachr. (199?) Let x B (x 0, δ) r 0. Then (x, t) K r L r for some t < t 1 if only if Γ(x, r) = 0, if this is the case, then t = f(x, r) = g(x, r). The functions f g are Lipschitz, thus Γ is Lipschitz, too. Therefore, the generalized subdifferential (gradient) Γ(x 0, r 0 ) of Γ at (x 0, r 0 ) IR n can be defined as in [4, p. 27]. Here, of course, we identify IR n with its dual space. Also note that Γ is defined in a full neighbourhood of (x 0, r 0 ), since we extended the geometric definition of f g. Claim 2. Let (ξ, η) Γ(x 0, r 0 ). Then ξ, u 0 γ := γ 1 γ 2 > 0; in particular, ( Γ(x 0, r 0 )) ({0} IR) =. Proof of Claim 2. Since f, g Γ are Lipschitz functions, they are differentiable on (B (x 0, δ) IR) A, where A IR n H n (A) = 0. Theorem in [4] then shows that { } Γ(x 0, r 0 ) = conv lim Γ(x i, r i ) (x i, r i ) (x 0, r 0 ), (x i, r i ) / A. i On the right-h side, we consider any sequence ((x i, r i )) i IN (B (x 0, δ) IR) A, converging to (x 0, r 0 ), such that the sequence ( Γ(x i, r i )) i IN converges. Then we take the convex hull of all such limit points. Therefore, it is sufficient to prove that ξ, u 0 γ is satisfied for all Let (ξ, η) { } lim Γ(x i, r i ) (x i, r i ) (x 0, r 0 ), (x i, r i ) / A. i (ξ, η) = lim i Γ(x i, r i ) = lim i ( f(x i, r i ) g(x i, r i )) for some (x i, r i ) (x 0, r 0 ) with (x i, r i ) / A. We write f(x i, r i ) = ( X f(x i, r i ), r f(x i, r i )), similarly for g. Taking subsequences, we may assume that X f(x i, r i ) ξ f X g(x i, r i ) ξ g for i. If h IR n 1, h < δ i is sufficiently large, then f(x i + h, r i ) f(x i, r i ) X f(x i, r i ), h g(x i + h, r i ) g(x i, r i ) X g(x i, r i ), h, due to the convexity of f(, r i ) g(, r i ). Passing to the limit, we obtain f(x 0 + h, r 0 ) f(x 0, r 0 ) ξ f, h g(x 0 + h, r 0 ) g(x 0, r 0 ) ξ g, h.

11 Hug/Schätzle, Intersections translative integral formulas 11 But since ˆf( ) = f(, r 0 ) ĝ( ) = g(, r 0 ) are convex, this demonstrates that ξ f ˆf(x 0 ) ξ g ĝ(x 0 ). Hence ξ = lim i ( X f(x i, r i ) X f(x i, r i )) = ξ f ξ g ˆf(x 0 ) ĝ(x 0 ). Now (2.6) yields that ξ, u 0 γ > 0. This completes the proof of Claim 2. After a rotation, we may assume that u 0 = e 1. Then Claim 2 yields that π 1 (ξ, η) γ > 0 for all (ξ, η) Γ(x 0, r 0 ), that is, π 1 Γ(x 0, r 0 ) has maximal rank in the sense of [4, pp. 253/256]. We also refer to [4] for a definition of the projection operator π 1. Write x 0 = (x 01, y 0 ) with x 01 IR y 0 IR n 2. According to Corollary in [4], there are neighbourhoods U(y 0, r 0 ) IR n 2 IR, U(x 01 ) IR a Lipschitz map ζ : U(y 0, r 0 ) U(x 01 ), satisfying ζ(y 0, r 0 ) = x 01, (ζ(y, r), y) B (x 0, δ) for all (y, r) U(y 0, r 0 ), such that Γ(ζ(y, r), y, r) = 0 for (y, r) U(y 0, r 0 ). Moreover, ζ(y, r) is the only solution of Γ(, y, r) = 0 on U(x 01 ) as long as (y, r) U(y 0, r 0 ). Let U(r 0 ) [0, 1] be a neighbourhood of r 0 in [0, 1] let U(y 0 ) IR n 2 be a neighbourhood of y 0 in IR n 2 such that U(y 0 ) U(r 0 ) U(y 0, r 0 ). If r 0 (0, 1), then we can choose U(r 0 ) = (r 1, r 2 ) [0, 1] set U := Kr 2 K r1. If r 0 = 0, then we choose U(r 0 ) = [0, r 2 ) [0, 1] set U := Kr 2, if r 0 = 1, then we choose U(r 0 ) = (r 1, 1] [0, 1] set U := IR n K r1. Then we obtain M U ((U(x 01 ) U(y 0 )) + (, 0)e n ) = {(x, t) K r L r (x, r) U(x 01 ) U(y 0 ) U(r 0 )} = {(ζ(y, r), y, f(ζ(y, r), y, r)) (y, r) U(y 0 ) U(r 0 )} = im h, where h : U(y 0 ) U(r 0 ) IR n is defined by h(y, r) := (ζ(y, r), y, f(ζ(y, r), y, r)). Clearly, h is Lipschitz the map q : im h U(y 0 ) U(r 0 ), defined by q(x 1, y, t) := (y, d(k, (x 1, y, t))), is Lipschitz, too, the inverse of h. This proves that M is an (n 1)-dimensional Lipschitz manifold with boundary M = S 0 S 1. Now let z 0 S r for some r 0. Certainly, it suffices to assume that r [0, 1]. Proceeding as before, we obtain that S r U ((U(x 01 ) U(y 0 )) + (, 0)e n ) = {(x, t) K r L r x U(x 01 ) U(y 0 )} = im h(, r),

12 12 Math. Nachr. (199?) where h(, r) : U(y 0 ) IR n is Lipschitz with inverse q : im h(, r) U(y 0 ), (x 1, y, t) y. Hence, S r is an (n 2)-dimensional Lipschitz manifold without boundary. Proposition 2.3. Under the assumptions (2.1), (1.9) (1.10), the set N, defined in (1.14), is an n-dimensional compact Lipschitz manifold with boundary N = M H 0 H 1 = H 0 H 1, 0 r 1 H r, for any r 0, is an (n 1)-dimensional compact Lipschitz manifold with boundary H r = S r. Proof. First, we show that N is an n-dimensional Lipschitz manifold with boundary N = M H 0 H 1. We continue using the notation results from the proof of Proposition 2.2. Let z 0 N. We distinguish several cases. Case 1: z 0 / M H 0 H 1. Then z 0 H r for some r (0, 1). Further, z 0 K r L r, since z 0 / M. Let an open neighbourhood V of z 0 s (0, r) be chosen in such a way that V L s V K 1 K s. Hence, z V implies that z K t for some t (s, 1) z L s L t. Thus, z K t L t = H t N. This shows that V N. Case 2: z 0 H 0 M. Then z 0 K L. There exist positive numbers δ > 0 r 0 > 0 such that W := (B(p 0, δ) + (, 0)e n ) K r 0 L, hence N W = {(y, f(y, r)) y B (x 0, δ), r [0, r 0 )}. Case 3: z 0 H 1 M. This case is analogous to Case 2. Case 4: z 0 M H 0 H 1 z 0 / (H 0 M) (H 1 M). This is equivalent to z 0 M. Thus, z 0 S r0 for some r 0 [0, 1]. We can write z 0 = (x 01, y 0, t 0 ) choose neighbourhoods U(x 01 ), U(y 0 ) U(r 0 ) as in the proof of Proposition 2.2. Certainly, we can assume that these neighbourhoods are open convex sets. Consider the function Γ(, y, r) = f(, y, r) g(, y, r), (y, r) U(y 0 ) U(r 0 ), defined on U(x 01 ). Recall from the proof of Proposition 2.2 that ζ(y, r) is the only solution of Γ(, y, r) = 0 on U(x 01 ) as long as (y, r) U(y 0 ) U(r 0 ). Hence, Γ(, y, r) is either strictly positive or strictly negative on each of the sets S r U + (y, r) := {s U(x 01 ) s > ζ(y, r)} U (y, r) := {s U(x 01 ) s < ζ(y, r)}.

13 Hug/Schätzle, Intersections translative integral formulas 13 Claim 1. If Γ(, y 0, r 0 ) > 0 on U + (y 0, r 0 ), then Γ(, y, r) > 0 on U + (y, r) for any (y, r) U(y 0 ) U(r 0 ). Proof of Claim 1. Assume that there is some (y 1, r 1 ) U(y 0 ) U(r 0 ) such that Γ(, y 1, r 1 ) < 0 on U + (y 1, r 1 ). Let ɛ > 0 be so small that for all s [0, 1]. Define ζ((1 s)y 0 + sy 1, (1 s)r 0 + sr 1 ) + ɛ U(x 01 ) ω(s) := (ζ((1 s)y 0 + sy 1, (1 s)r 0 + sr 1 ) + ɛ, (1 s)y 0 + sy 1, (1 s)r 0 + sr 1 ), where s [0, 1]. Then ω(s) U(x 01 ) U(y 0 ) U(r 0 ), for s [0, 1], Γ(ω(0)) > 0 Γ(ω(1)) < 0. Hence there is some s [0, 1] for which Γ(ω(s )) = 0. But this is a contradiction, since ɛ > 0, which completes the proof of Claim 1. Of course, similar statements hold under the assumption Γ(, y 0, r 0 ) < 0 or with U (y 0, r 0 ) instead of U + (y 0, r 0 ). Claim 2. Precisely one of the following two conditions is fulfilled. (i) Γ(, y, r) > 0 on U + (y, r) Γ(, y, r) < 0 on U (y, r), for all (y, r) U(y 0 ) U(r 0 ). (ii) Γ(, y, r) < 0 on U + (y, r) Γ(, y, r) > 0 on U (y, r), for all (y, r) U(y 0 ) U(r 0 ). Proof of Claim 2. Assume that neither (i) nor (ii) is satisfied. Then according to Claim 1 we can infer that either Γ(s, y, r) 0 for all (s, y, r) U(x 01 ) U(y 0 ) U(r 0 ) or Γ(s, y, r) 0 for all (s, y, r) U(x 01 ) U(y 0 ) U(r 0 ). But then Γ has a local minimum or a local maximum at (x 01, y 0, r 0 ) = (x 0, r 0 ). From Proposition in [4] we then obtain that 0 Γ(x 0, r 0 ), which contradicts Claim 2 in the proof of Proposition 2.2. This proves Claim 2. Assume, for example, that (i) is satisfied. Then {(s, y, f(s, y, r)) (s, y, r) U(x 01 ) U(y 0 ) U(r 0 ), s ζ(y, r)} K r L r {(s, y, f(s, y, r)) (s, y, r) U(x 01 ) U(y 0 ) U(r 0 ), s < ζ(y, r)} ( K r L r ) =. Defining the open set U as in the proof of Proposition 2.2, we now see that N U ((U(x 01 ) U(y 0 )) + (, 0)e n ) = {(x, t) K r L r (x, r) U(x 01 ) U(y 0 ) U(r 0 )} = {(s, y, f(s, y, r)) (s, y, r) U(x 01 ) U(y 0 ) U(r 0 ), s ζ(y, r)}. The case where (ii) is satisfied can be treated similarly.

14 14 Math. Nachr. (199?) If we collect the results that have been deduced in the preceding four cases, then we see that N enjoys the asserted properties. Next we prove that H r, r [0, 1], is an (n 1)-dimensional Lipschitz manifold with boundary H r = S r. Let r be fixed z 0 H r. Case 1: z 0 H r S r. Then z 0 K r L r. Let δ > 0 be sufficiently small so that Then W := (B(p 0, δ) + (, 0)e n ) K r L r. H r W = {(x, f(x, r) x B (x 0, δ) }. Case 2: z 0 S r. Using the same arguments as in Case 4 above under the assumption that, for instance, condition (i) of Claim 2 above is satisfied, we obtain H r U ((U(x 01 ) U(y 0 )) + (, 0)e n ) = {(x, t) K r L r x U(x 01 ) U(y 0 )} = {(s, y, f(s, y, r)) (s, y) U(x 01 ) U(y 0 ), s ζ(y, r)}. From the representations derived in these two cases the assertion follows immediately A Morse type lemma In this subsection, we apply elementary techniques from Morse theory to show that (2.7) S r = Ss for all r, s > 0. We write d K (x) := d(k, x) for the distance of x IR n to K. Lipschitz function. In IR n K, it is known that d K C 1 Clearly, d K ( ) is a ν K (x) := d K (x) = x p K(x) d K (x) is the outer unit normal at x K r of the parallel body K r, r = d K (x), where p K (x) denotes the unique next point of K to x. Likewise, we define d L for L. We see that (2.8) d K C 1,1 loc ([d K > 0]) C 0,1 (IR n ), d L C 1,1 loc ([d L > 0]) C 0,1 (IR n ). Moreover, we observe that K r = [d K = r] L r = [d L = r] for r > 0. As in Morse theory, we consider a flow which is induced by (2.9) F (x) := ν K + ν L (x) for x V, 1 + ν K, ν L where V := IR n (K L) = [d K > 0, d L > 0]. The flow F is well-defined on V, that is, 1 + ν K, ν L > 0 on V. Indeed, if there is x V which satisfies d K (x) = r > 0,

15 Hug/Schätzle, Intersections translative integral formulas 15 d L (x) = s > ν K (x), ν L (x) = 0, then we infer from ν K (x) = ν L (x) = 1 that ν K (x) = ν L (x). As ν K (x) N(K r, x) ν L (x) N(L s, x) for r, s > 0, this contradicts (2.2). Obviously, F satisfies F, ν K = F, ν L = 1. Observing the continuity properties of ν K = d K, ν L = d L given in (2.8), we obtain (2.10) F C 0,1 loc (V ). Hence there is a unique local solution of the ordinary differential equation (2.11) x (t) = F (x(t)), x(r) = x 0, for x 0 S r, on some maximal interval I x0 containing r. We compute d dt d K(x(t)) = ν K (x(t)), F (x(t)) = 1, likewise for L. We obtain d K (x(t)) = d L (x(t)) = t, hence x(t) S t for t I x0. Since x(t) approaches the boundary of V when t approaches an endpoint of I x0, we conclude that = (0, ). I x0 Taking any x 0 S 1, we obtain a continuous map (2.12) Φ : S 1 (0, ) r>0 S r such that Φ(x 0, t) = x(t) S t. Since the solution of (2.11) is unique, we see that Φ is bijective, the theory of ordinary differential equations implies that Φ is even a homeomorphism. We put ϕ t (x) := Φ(x, t) see that ϕ t induces a homeomorphism (2.13) ϕ t : S 1 S t for t > 0. Next we want to extend the map Φ to t = 0 in order to get a continuous, surjective map ϕ : S 1 S 0. To this end, we need a uniform Lipschitz bound on the solutions of (2.11) for small t > 0, that is a bound on F. The next lemma implies that (2.14) F C on [0 < d K, d L 1], for some 0 < C <. Lemma 2.4. There is a constant c 0 > 0 such that (2.15) 1 + ν K (x), ν L (x) c 0 for x [0 < d K, d L 1].

16 16 Math. Nachr. (199?) Proof. If there is no such c 0, then there are points x j IR n, j IN, such that 0 < d K (x j ), d L (x j ) ν K (x j ), ν L (x j ) 0 for j. As K 1, L 1 are compact ν K = ν L = 1, we may assume, by passing to a subsequence, that x j x 0, d K (x 0 ) = r [0, 1], d L (x 0 ) = s [0, 1], ν K (x j ) ξ K, ν L (x j ) ξ L, as j. We infer that x 0 K r L s, ξ K = ξ L = 1, 1 + ξ K, ξ L = 0, hence ξ K = ξ L. As ν K, ν L are the outer normals at parallel bodies of the convex bodies K, L, respectively, we conclude that ξ K N(K r, x 0 ) ξ L N(L s, x 0 ). As ξ K = ξ L, this contradicts (2.2). Relation (2.14) implies that for x S 1 0 < s, t 1. We define Φ(x, t) Φ(x, s) C t s Φ(x, 0) := ϕ(x) := lim t 0 Φ(x, t), as the limit exists. Since d K (Φ(x, t)) = d L (Φ(x, t)) = t, we obtain Φ(x, 0) = ϕ(x) S 0. Some properties of Φ ϕ, which we need subsequently, are stated in the next lemma. Lemma 2.5. The maps Φ : S 1 [0, 1] M = are continuous surjective. 0 r 1 Proof. For x, y S 1 0 < t 1, we obtain S r ϕ : S 1 S 0 ϕ(x) ϕ(y) ϕ(x) Φ(x, t) + ϕ t (x) ϕ t (y) + Φ(y, t) ϕ(y) 2C t + ω t ( x y ), where ω t is a modulus of continuity for the continuous function ϕ t. As t > 0 can be chosen arbitrarily small, we infer that ϕ is continuous. Next we consider (x j, t j ) (x 0, 0) for x j, x 0 S 1. We obtain Φ(x j, t j ) Φ(x 0, 0) Φ(x j, t j ) Φ(x j, 0) + ϕ(x j ) ϕ(x 0 ) C t j + ω( x j x 0 ),

17 Hug/Schätzle, Intersections translative integral formulas 17 where ω is a modulus of continuity for the continuous function ϕ. As t j 0 x j x 0, we see that Φ(x j, t j ) Φ(x 0, 0), thus Φ is continuous. To prove surjectivity, we choose x 0 S 0. By Proposition 2.2(a), M is a topological manifold with boundary M = S 0 S 1. Hence, there exist x j M M = 0<r<1 S r, j IN, such that x j x 0 as j. As Φ : S 1 (0, ) r>0 S r is surjective, for each j IN there are x j S 1 r j > 0 such that Φ( x j, r j ) = x j. We infer that r j 0, as S 1 is compact, passing to a subsequence (if necessary) we may assume that x j x S 1. As Φ is continuous according to the first part of the proof, we obtain ϕ( x) = Φ( x, 0) Φ( x j, t j ) = x j x 0 as j, therefore Φ ϕ are surjective. As continuous, surjective maps from compact sets are identification or quotient maps (see [6, VI.1.4 XI.2.1]), the lemma yields that Φ ϕ are quotient maps. Since the set S 1 [0, 1] [0, 1] is compact, we obtain that Φ id : S 1 [0, 1] [0, 1] M [0, 1] is a quotient map, too. That Φ id is a quotient map for a quotient map Φ follows by a theorem of Whitehead (see [6, XII.4.1]), even without using that S 1 [0, 1] [0, 1] is compact. We use these quotient maps in the next proposition to construct a homotopy which will show that S 0 is a strong deformation retract of M. Proposition 2.6. The set S 0 is a strong deformation retract of M, hence these sets are of the same homotopy type S 0 M. Proof. Clearly, S 1 {0} is a strong deformation retract of S 1 [0, 1] via the homotopy H : S 1 [0, 1] [0, 1] S 1 [0, 1] defined by We consider the commutative diagram H(x, t, s) := (x, t(1 s)).

18 18 Math. Nachr. (199?) H S 1 [0, 1] [0, 1] S 1 [0, 1] Φ id M [0, 1] H Φ M As H preserves the relations induced by Φ id Φ, that is [(Φ id)(x, t, s) = (Φ id)(x, t, s )] [(Φ H)(x, t, s) = (Φ H)(x, t, s )], passing to the quotient (see [6, I.7.7]), we obtain a map H : M [0, 1] M such that the diagram commutes. Further, H is continuous by [6, VI.4.3], as Φ id is a quotient map. The map H is the desired strong deformation retraction for S 0. Indeed, H(x, 0) = x for x M, H(y, s) = y for y S 0, s [0, 1], H(x, 1) S 0 for x M. These assertions are easily verified from the definition of H. Since it has already been proved that Φ is a homeomorphism, the following lemma is much easier obtained than Proposition 2.6. Lemma 2.7. The set S 1 is a strong deformation retract of M M, hence these 2 sets are of the same homotopy type S 1 2 M M. Proof. Clearly, S 1 { 1 2 } is a strong deformation retract of S 1 (0, 1). As Φ : ( S 1 (0, 1), S 1 { 1 2 }) ( 0<r<1 S r, S 1 2 induces a homeomorphism (see (2.12)), we obtain that S 1 is a strong deformation 2 retract of 0<r<1 S r. According to Proposition 2.2(a), we know that 0<r<1 S r = M M, which concludes the proof. From (2.13) we now obtain Proposition 2.8. The sets S r M M are of the same homotopy type, that is S r M M for r > 0. )

19 Hug/Schätzle, Intersections translative integral formulas Collaring Theorem 1.3 will follow from Propositions when we know that (2.16) i : M M M is a homotopy equivalence, that is M M is a weak deformation retract of M. In [30, p. 297], (2.16) is mentioned as an easy consequence of the following theorem due to Brown [3, Theorem 2]. Theorem (Brown). The boundary of a topological manifold with boundary is collared. A subset B M is called collared in M if there is a homeomorphism h : (B [0, 1), B {0}) (U, B) onto some open neighbourhood U of B; see [3, II]. It is immediate that the boundary of a manifold is locally collared, Brown s Theorem states that the boundary has a global collaring. For the sake of completeness, we give the arguments for proving (2.16) with the use of Brown s Theorem. Corollary 2.9. Let Y be a compact topological manifold with boundary Y. Then Y Y is a weak deformation retract of Y. Proof. According to Brown s Theorem, we may identify an open neighbourhood U Y of Y with Y [0, 1), that is we assume that U = Y [0, 1) Y is an open neighbourhood of Y = Y {0}. We put Z := Y ( Y [0, 1 2 )) which is closed as Y [0, 1 2 ) is open, Z := Y ( Y [0, 1 2 ]) which is open as Y Y [0, 1 2 ] are compact, hence are closed in Y. We claim that Z is a strong deformation retract of Y. To this end, we define a homotopy H : Y [0, 1] Y by putting H(y, s) := { y if y Z, (ỹ, t s min{0, t 1 2 }) if y = (ỹ, t) U = Y [0, 1). The map H is well-defined continuous, as Y = Z U Z U are open. Moreover, H(y, 0) = y for y Y, H(z, s) = z for z Z, s [0, 1], H(y, 1) Z for y Y,

20 20 Math. Nachr. (199?) which yields that Z is a strong deformation retract of Y. We observe that H : (Y Y ) [0, 1] Y Y, Z Y Y is a strong deformation retract of Y Y. Therefore the inclusion maps on the left side in the commutative diagram Z Y Y i Y are homotopy equivalences, hence so is i, thus Y Y is a weak deformation retract of Y. As already mentioned, Theorem 1.3 is a consequence of (2.16) the Propositions Intersection of boundary body In this subsection, we supplement the details for the proof of Theorem 1.4. We use the notation of Section 2.2. First, we modify the flow defined in (2.9) to prove that (2.17) H r = Hs for all r, s > 0. The flow F was defined in V = [d K > 0, d L > 0]. It transforms S r into S s. Now we are looking for a flow F which transforms H r into H s. We recall that S r = [d K = d L = r] H r = [d K = r, d L r] for r > 0. As H r = S r, according to Proposition 2.3, we take F = F on [d K = d L > 0] extend it on Ṽ := [d K > 0]. More precisely, we put (2.18) ν K + ν L F (x) := η(d K (x), d L (x)) 1 + ν K, ν L (x) + (1 η(d K(x), d L (x)))ν K (x), for x Ṽ, where η : (0, ) IR [0, 1], η C0,1 loc ((0, ) IR), is such that η(r, r) = 1 for r > 0, η(r, s) = 0 for r > 0, s < r 2. For example, we may choose η(r, s) := min { r 1 max{2s r, 0}, 1 } for r > 0.

21 Hug/Schätzle, Intersections translative integral formulas 21 Observing (2.8), (2.10), the properties of η, we see that F C 0,1 loc (Ṽ ). Moreover, we calculate that F, ν K = 1 on Ṽ F, ν L = 1 on r>0 S r. As in Section 2.2, there is a unique local solution of the ordinary differential equation (2.19) x (t) = F (x(t)), x(r) = x 0, for x 0 K r Ṽ, on some maximal interval J x 0 containing r > 0. We compute hence d dt d K(x(t)) = ν K (x(t)), F (x(t)) = 1, x(t) K t for t J x0. Furthermore, J x0 = (0, ) since x(t) approaches the boundary of Ṽ = [d K > 0] when t approaches an endpoint of J x0. Since the solution of the ordinary differential equation (2.19) is unique, we see that the flow F establishes a homeomorphism Ψ r,s : K r = Ks for r, s > 0 which maps x(r) = x 0 to x(s). Next, when x 0 S r K r, we consider the solution x of (2.11) observe, as x(t) S t F = F on r>0 S r, that x is the solution of (2.19) as well. Therefore Ψ r,s (S r ) = S s. This yields (2.20) Ψ r,s (H r ) = H s for r, s > 0. Indeed, H r = K r [d L r]. Let x 0 H r. If d L (x 0 ) = r, then x 0 S r x(t) S t H t for all t > 0. If d L (x 0 ) < r, then x 0 S r x(t) S t, as Ψ r,t (S r ) = S t Ψ r,t : K r = Kt is bijective. Therefore d L (x(t)) t for all t > 0. As d L (x(r)) < r, we see that d L (x(t)) < t for all t > 0, x(t) H t. Hence Ψ r,s (H r ) H s, equality is concluded by symmetry in r, s. Clearly, (2.20) implies (2.17). Taking any x 0 H 1, we obtain a continuous map ( ) (2.21) Ψ : (H 1, S 1 ) (0, ) S r r>0 H r, r>0 such that Ψ(x 0, t) = x(t) H t. As the solutions of (2.19) are unique, Ψ is a homeomorphism. We put ψ t (x) := Ψ(x, t), thus ψ t induces a homeomorphism (2.22) ψ t : (H 1, S 1 ) (H t, S t ) for t > 0.

22 22 Math. Nachr. (199?) We want to extend Ψ to t = 0. Lemma 2.4 shows that F C on [0 < d K 1, d L 1], Ψ admits a uniform Lipschitz constant in the variable t. As in Lemma 2.5, we can extend Ψ ψ t to obtain continuous surjective maps, hence quotient maps, Ψ : H 1 [0, 1] N = 0 r 1 H r ψ : H 1 H 0. The surjectivity is obtained by using that H 0 N N N; see Proposition 2.3. Next, using these quotient maps, we conclude as in Lemma 2.6 that H 0 is a strong deformation retract of N, in particular (2.23) H 0 N. We recall that H 1 = S 1 N = ( 2 2 0<r<1 S r) H 0 H 1, see Proposition 2.3. Knowing that Ψ : (H 1, S 1 ) (0, 1) ( 0<r<1 H r, 0<r<1 S r) is a homeomorphism, we see as in Lemma 2.7 that H 1 H 1 is a strong deformation retract of N N. 2 2 In particular, (2.24) H 1 2 H 1 2 N N. Finally, the proof of Theorem 1.4 is concluded again by Brown s Theorem its Corollary 2.9, which yields that (2.25) H 1 2 H 1 2 H 1 2 N N N. Then (2.17), (2.23), (2.24) (2.25) together imply which is the assertion of Theorem 1.4. H r H 0 for r 0, 3. The translative integral formulas In this final section, we shall prove Theorems The basic idea of our approach has already been outlined in the Introduction. The following crucial lemma is implicitly contained in a paper of Ewald, Larman Rogers [7]. It is also used by Heinrich Molchanov [16]. For the reader s convenience we include the complete argument. Subsequently, we write H r, r 0, for the r-dimensional Hausdorff measure in a Euclidean space. It will always be clear from the context which space is meant. Lemma 3.1. Let K, L IR n be convex bodies. Then the set T (K, L) of all t IR n such that K L + t do not intersect almost transversally has H n measure zero.

23 Hug/Schätzle, Intersections translative integral formulas 23 Proof. Certainly, T (K, L) = T 1 (K, L) T 2 (K, L), where T 1 (K, L) denotes the set of all t IR n for which there is some x K (L + t) such that N(K, x) N(L + t, x) {0} T 2 (K, L) is the set of all t IR n for which there is some x K (L + t) such that N(K, x) ( N(L + t, x)) {0}. It is easy to check that T 2 (K, L) = (K + ( L)). For a convex body M IR n u S n 1, let F (M, u) denote the support set of M with exterior normal vector u, let h(m, ) : IR n IR denote the support function of M; see [25] for explicit definitions. Choose any t T 1 (K, L). Then there exists a point x K (L + t) a unit vector u N(K, x) N(L + t, x), thus x, u = h(k, u) x t, u = h(l, u). Then t = x (x t) K + ( L) t F (K, u) + ( F (L, u)). This shows that T 1 (K, L) [F (K, u) + ( F (L, u))] =: Λ(L, K). u S n 1 Let ɛ > 0 be arbitrarily chosen. Then Theorem in [25] implies that there are caps C 1,..., C m whose union covers (K + L) for which m H n (C i ) c 1 ɛ, i=1 where c 1 is a constant independent of ɛ. Further, Lemma in [25] yields that Λ(L, K) m [C i + ( C i ) + a i ], where a 1,..., a m are suitable translation vectors. This implies i=1 i=1 m m H n (Λ(L, K)) H n (C i + ( C i )) (n + 1) n H n (C i ) c 2 ɛ, where c 2 is a constant independent of ɛ. This concludes the proof since ɛ > 0 was arbitrary. Using Lemma 3.1 the continuity properties of mixed volumes, we shall see that Theorems follow from the next lemma. Lemma 3.2. Let K, L IR n be convex bodies, let r > 0. Define X := K r Y := L r. Then Theorems hold for the parallel bodies X Y. Proof. First we note that X Y are sets of positive reach in the sense of Federer [8]. In fact, these sets are C 1,1 Lipschitz submanifolds of IR n. Of course, the i=1

24 24 Math. Nachr. (199?) same is true for X Y. For general sets of positive reach, Federer [8] introduced curvature measures as Borel measures on IR n. Subsequently, M. Zähle [36] extended this concept to (generalized) curvature measures, as Borel measures on IR n IR n, simplified the original approach. For a set R IR n of positive reach, these measures are denoted by C i (R, ), i = 0,..., n 1. They can be obtained as coefficient measures of a local Steiner formula. We refer to [36] to [19] for a more detailed description of these measures; see also [36] [19] for the particular normalization of these measures which is adopted in the present paper. Next we note that if X (Y + t) t / T 2 (X, Y ), then X ( Y + t) is a set of positive reach (3.1) χ( X ( Y + t)) = C 0 ( X ( Y + t), IR n IR n ). This is implied by Theorems in [8]. Hence it follows from results provided in [8] that t χ( X ( Y + t)) is Borel measurable if we possibly redefine this map on the Borel set T 2 (X, Y ) of measure zero where both sides of equation (3.1) might not be defined. Essentially, Lemma 3.2 will be proved by applying a translative integral formula, due to Rataj M. Zähle [19], in the present special context. Since we shall need some specific information about certain mixed curvature measures which are involved in the statement of this formula, we introduce some additional notation. Let ν X (x), x X, denote the uniquely determined exterior unit normal vector of X at x. Let N(X) N( X) denote the generalized unit normal bundles of X X, respectively. A general definition is given, e.g., in [36]. In the present particular setting, N(X) = {(x, ν X (x)) IR n S n 1 : x X} N( X) = N(X) {(x, ν X (x)) IR n S n 1 : x X}. Since X = K r r > 0, the map ν X is Lipschitz hence differentiable H n 1 - almost everywhere on X. Thus, by specializing the general definitions in [36], the generalized curvature functions κ 1 (X, x, m),..., κ n 1 (X, x, m) [0, ) can be obtained, for H n 1 -almost all x X m = ν X (x), as the eigenvalues of the symmetric linear map Dν X (x) : ν X (x) ν X (x). Moreover, for H n 1 -almost all (x, m) N(X) i = 1,..., n 1, κ i ( X, x, m) = κ i (X, x, m) [0, ) κ i ( X, x, m) = κ i (X, x, m) (, 0]. A common eigenvector corresponding to κ i ( X, x, m), κ i ( X, x, m) κ i (X, x, m) is denoted by a i ( X, x, ±m) = a i (X, x, m), for i = 1,..., n 1 H n 1 -almost all (x, m) N(X). In particular, these eigenvectors are chosen in such a way that they represent an orthonormal basis of m. Similar results notations are used for Y. All these definitions are consistent with those given in [36] for arbitrary sets of positive

25 Hug/Schätzle, Intersections translative integral formulas 25 reach. Finally, for vectors c 1,..., c n IR n the volume of the parallelepiped spanned by these vectors is denoted by c 1... c n, since this volume equals the norm of the n-vector c 1... c n. After these preparations we can describe the mixed curvature measures Ψ l,n l (R 1, R 2 ; ), l {0,..., n}, where R 1, R 2 {X, Y, X, Y }. These measures are Borel measures on subsets of (IR n ) 3. In [19], these measures are defined by an integration over the joint unit normal bundle of R 1 R 2 (that is, by means of a joint unit normal current of R 1 R 2 ), for general sets R 1, R 2 of positive reach. We shall merely need the projections of the mixed curvature measures onto the first two components, hence we set Ψ l,n l (R 1, R 2 ; A) := Ψ l,n l (R 1, R 2 ; A IR n ), for an arbitrary Borel set A IR n IR n. By Theorem 2 in [19], for l = 1,..., n 1 the measures Ψ l,n l (R 1, R 2 ; ) can be represented as Ψ l,n l (R 1, R 2 ; A) = 1 A (x, y)f l,n l (θ(m, n)) N(R 1) N(R 2) ( n 1 i=1 1 + κi (R 1, x, m) κ i (R 2, y, n) 2 ) 1 κ i (R 1, x, m) κ j (R 2, y, n) I =l J =n l i I c j J c a i (R 1, x, m) 2 a j (R 2, y, n) H 2n 2 (d(x, m, y, n)), i I j J where θ(m, n) (0, π) denotes the angle between two vectors m, n S n 1 m ±n with F l,n l (θ) = (H n 1 (S n 1 )) 1 θ sin θ 1 0 ( sin(tθ) sin θ ) l 1 ( ) n 1 l sin((1 t)θ) dt, sin θ θ (0, π). It should be emphasized that for m = ±n I = n J we obtain a i (R 1, x, m) a j (R 2, y, n) i I j J = 0, since then all vectors lie in the orthogonal complement of m. So if m = ±n, then the integr is defined to be zero. We also mention that (3.2) (3.3) Ψ 0,n (R 1, R 2 ; A B) = C 0 (R 1, A IR n )H n (R 2 B) Ψ n,0 (R 1, R 2 ; A B) = H n (R 1 A)C 0 (R 2, B IR n ),

26 26 Math. Nachr. (199?) for Borel sets A, B IR n. By Lemma 3.1, condition (6) of Theorem 1 in [19] is satisfied for the sets X Y. Thus we can apply this theorem to the sets X Y in IR n ( for k = 0 in the notation of [19]). Let h : IR n [0, ) be an arbitrary Borel measurable function. Furthermore, let B (IR n ) 3 denote a bounded Borel set which contains (X + ( Y )) X S n 1 in its interior. If we use the fact that the curvature measure C 0 ( X ( Y + t), ) is defined at least for H n -almost all t IR n such that X ( Y +t) concentrated on X S n 1, X ( Y +t) = if t / X +( Y ), Ψ l,n l ( X, Y ; ) is concentrated on X Y S n 1 for l = 0,..., n, equation (3.1), then an application of Theorem 1 in [19] to the function 1 B h yields (3.4) h(t)χ( X ( Y + t))h n (dt) IR n n = h(x y)ψ l,n l ( X, Y ; d(x, y)). (IR n ) 2 l=0 Since H n ( X) = H n ( Y ) = 0, equations (3.2) (3.3) imply that the summs corresponding to l = 0 l = n vanish. Next we consider the following decomposition of the essential domain of integration for each summ I(h, l), l = 1,..., n 1, on the right-h side of equation (3.4): N( X) N( Y ) = (N(X) N(Y )) ({(x, m) : (x, m) N(X)} N(Y )) (N(X) {(y, n) : (y, n) N(Y )}) ({(x, m) : (x, m) N(X)} {(y, n) : (y, n) N(Y )}). This decomposition leads to a corresponding decomposition of I(h, l) according to (3.5) I(h, l) = I 1 (h, l) I 4 (h, l), l = 1,..., n 1. Thus, for l = 1,..., n 1 we obtain I 1 (h, l) = h(x y)ψ l,n l (X, Y ; d(x, y)). (IR n ) 2 In order to treat the second summ I 2 (h, l), we first apply the isometric map (x, m, y, n) ( x, m, y, n) then use the relations κ i ( X, x, m) = κ i ( X, x, m) = κ i ( X, x, m) a i ( X, x, m) = ±a i ( X, x, m) = ±a i ( X, x, m), for i = 1,..., n 1 H n 1 -almost all (x, m) N( X). Thus we obtain Similarly, we find I 2 (h, l) = ( 1) n 1 l (IR n ) 2 h( x y)ψ l,n l ( X, Y ; d(x, y)). I 3 (h, l) = ( 1) l 1 (IR n ) 2 h(x + y)ψ l,n l (X, Y ; d(x, y))