In this final section, we shall prove Theorems 1.1 and 1.2. The basic idea of our approach has already been outlined in the Introduction.

The following crucial lemma is implicitly contained in a paper ofEwald,Larman
andRogers[7]. It is also used byHeinrichandMolchanov[16]. For the reader’s
convenience we include the complete argument. Subsequently, we write H^{r}, r ≥ 0,
for ther-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 allt∈IR^{n}
such thatK andL+tdo not intersect almost transversally has H^{n} measure zero.

P r o o f . Certainly, T(K, L) =T1(K, L)∪T2(K, L), whereT1(K, L) denotes the set
of allt∈IR^{n} for which there is somex∈∂K∩∂(L+t) such that

N(K, x)∩N(L+t, x)6={0}

andT2(K, L) is the set of allt∈IR^{n} for which there is somex∈∂K∩∂(L+t) such
that

N(K, x)∩(−N(L+t, x))6={0}.

It is easy to check thatT2(K, L) =∂(K+ (−L)).

For a convex bodyM ⊂IR^{n}andu∈S^{n−1}, letF(M, u) denote the support set ofM
with exterior normal vectoru, and let h(M,·) : IR^{n}→IR denote the support function
ofM; see [25] for explicit definitions. Choose anyt ∈T1(K, L). Then there exists a
pointx∈∂K∩∂(L+t) and a unit vectoru∈N(K, x)∩N(L+t, x), and thus

hx, ui=h(K, u) and hx−t, ui=h(L, u).

Thent=x−(x−t)∈∂K+ (−∂L) andt∈F(K, u) + (−F(L, u)). This shows that
T_{1}(K, L)⊆ [

u∈S^{n−1}

[F(K, u) + (−F(L, u))] =: Λ(L, K).

Let > 0 be arbitrarily chosen. Then Theorem 2.3.2 in [25] implies that there are capsC1, . . . , Cmwhose union covers ∂(K+L) and for which

m

X

i=1

H^{n}(Ci)≤c1,

wherec_{1} is a constant independent of. Further, Lemma 2.3.9 in [25] yields that
Λ(L, K)⊆

m

[

i=1

[C_{i}+ (−C_{i}) +a_{i}],
wherea_{1}, . . . , a_{m} are suitable translation vectors. This implies

H^{n}(Λ(L, K))≤

m

X

i=1

H^{n}(C_{i}+ (−C_{i}))≤

m

X

i=1

(n+ 1)^{n}H^{n}(C_{i})≤c_{2},

where c_{2} is a constant independent of . This concludes the proof since > 0 was

arbitrary. 2

Using Lemma 3.1 and the continuity properties of mixed volumes, we shall see that Theorems 1.1 and 1.2 follow from the next lemma.

Lemma 3.2. Let K, L⊂IR^{n} be convex bodies, and letr >0. DefineX :=Kr and
Y :=Lr. Then Theorems 1.1 and 1.2 hold for the parallel bodies X andY.

P r o o f . First we note that ∂X and ∂Y are sets of positive reach in the sense of
Federer[8]. In fact, these sets areC^{1,1}Lipschitz submanifolds of IR^{n}. Of course, the

same is true forX andY. For general sets of positive reach,Federer[8] introduced
curvature measures as Borel measures on IR^{n}. Subsequently,M. Z¨ahle[36] extended
this concept to (generalized) curvature measures, as Borel measures on IR^{n}×IR^{n}, and
simplified the original approach. For a setR⊂IR^{n} of positive reach, these measures
are denoted byCi(R,·),i= 0, . . . , n−1. They can be obtained as coefficient measures
of a local Steiner formula. We refer to [36] and to [19] for a more detailed description
of these measures; see also [36] and [19] for the particular normalization of these
measures which is adopted in the present paper. Next we note that ifX∩(Y +t)6=∅
andt /∈T2(X, Y), then∂X∩(∂Y +t) is a set of positive reach and

χ(∂X∩(∂Y +t)) =C_{0}(∂X∩(∂Y +t),IR^{n}×IR^{n}).

(3.1)

This is implied by Theorems 4.10 and 5.19 in [8]. Hence it follows from results provided
in [8] thatt7→χ(∂X∩(∂Y +t)) is Borel measurable if we possibly redefine this map
on the Borel setT_{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 toRatajandM. Z¨ahle[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.

LetνX(x), x∈∂X, denote the uniquely determined exterior unit normal vector of X atx. LetN(X) andN(∂X) denote the generalized unit normal bundles of X and

∂X, respectively. A general definition is given, e.g., in [36]. In the present particular setting,

N(X) ={(x, νX(x))∈IR^{n}×S^{n−1}:x∈∂X}

and

N(∂X) =N(X)∪ {(x,−νX(x))∈IR^{n}×S^{n−1}:x∈∂X}.

Since X = K_{r} and r > 0, the map ν_{X} is Lipschitz and 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 allx∈∂X and m=νX(x), as the eigenvalues of
the symmetric linear mapDνX(x) : νX(x)^{⊥} →νX(x)^{⊥}. Moreover, for H^{n−1}-almost
all (x, m)∈N(X) andi= 1, . . . , n−1,

κi(∂X, x, m) =κi(X, x, m)∈[0,∞) and

κi(∂X, x,−m) =−κi(X, x, m)∈(−∞,0].

A common eigenvector corresponding toκi(∂X, x, m),κi(∂X, x,−m) andκi(X, x, m)
is denoted byai(∂X, x,±m) = ai(X, x, m), for i= 1, . . . , n−1 and 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 ofm^{⊥}. Similar results and notations are used for Y.
All these definitions are consistent with those given in [36] for arbitrary sets of positive

reach. Finally, for vectorsc1, . . . , cn ∈IR^{n} the volume of the parallelepiped spanned
by these vectors is denoted bykc1∧. . .∧cnk, since this volume equals the norm of the
n-vectorc1∧. . .∧cn.

After these preparations we can describe the mixed curvature measures Ψl,n−l(R1, R2;·), 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 ofR_{1} andR_{2} (that is, by means of a joint unit normal current ofR_{1} andR_{2}),
for general setsR_{1}, R_{2} of positive reach. We shall merely need the projections of the
mixed curvature measures onto the first two components, and hence we set

Ψ_{l,n−l}(R1, R2;A) := Ψ_{l,n−l}(R1, R2;A×IR^{n}),

since then all vectors lie in the orthogonal complement ofm. So if m=±n, then the integrand is defined to be zero. We also mention that

Ψ0,n(R1, R2;A×B) =C0(R1, A×IR^{n})H^{n}(R2∩B)
(3.2)

and

Ψ_{n,0}(R_{1}, R_{2};A×B) =H^{n}(R_{1}∩A)C_{0}(R_{2}, B×IR^{n}),
(3.3)

for Borel setsA, B⊆IR^{n}. By Lemma 3.1, condition (6) of Theorem 1 in [19] is satisfied
for the sets∂X and ∂Y. Thus we can apply this theorem to the sets ∂X and ∂Y in
IR^{n} (and fork= 0 in the notation of [19]). Leth: IR^{n}→[0,∞) be an arbitrary Borel
measurable function. Furthermore, letB ⊂(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
measureC0(∂X∩(∂Y +t),·) is defined at least for H^{n}-almost allt∈IR^{n} such that

∂X∩(∂Y+t)6=∅and concentrated on∂X×S^{n−1},∂X∩(∂Y +t) =∅ift /∈X+ (−Y),
Ψl,n−l(∂X, ∂Y;·) is concentrated on ∂X×∂Y ×S^{n−1} forl = 0, . . . , n, and equation
(3.1), then an application of Theorem 1 in [19] to the function1Bhyields

Z

IR^{n}

h(t)χ(∂X∩(∂Y +t))H^{n}(dt)
(3.4)

=

n

X

l=0

Z

(IR^{n})^{2}

h(x−y)Ψ_{l,n−l}(∂X, ∂Y;d(x, y)).

Since H^{n}(∂X) = H^{n}(∂Y) = 0, equations (3.2) and (3.3) imply that the summands
corresponding tol= 0 andl=nvanish.

Next we consider the following decomposition of the essential domain of integration for each summandI(h, l),l= 1, . . . , n−1, on the right-hand 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 ofI(h, l) according to I(h, l) =I1(h, l) +. . .+I4(h, l), l= 1, . . . , n−1.

(3.5)

Thus, forl= 1, . . . , n−1 we obtain
I_{1}(h, l) =

Z

(IR^{n})^{2}

h(x−y)Ψ_{l,n−l}(X, Y;d(x, y)).

In order to treat the second summand I2(h, l), we first apply the isometric map (x, m, y, n)7→(−x, m, y, n) and then use the relations

κi(∂X,−x, m) =−κi(−∂X, x, m) =−κi(−X, x, m) and

a_{i}(∂X,−x, m) =±a_{i}(−∂X, x, m) =±a_{i}(−X, x, m),
fori= 1, . . . , n−1 andH^{n−1}-almost all (x, m)∈N(−X). Thus we obtain

I_{2}(h, l) = (−1)^{n−1−l}
Z

(IR^{n})^{2}

h(−x−y)Ψ_{l,n−l}(−X, Y;d(x, y)).

Similarly, we find

I3(h, l) = (−1)^{l−1}
Z

(IR^{n})^{2}

h(x+y)Ψ_{l,n−l}(X,−Y;d(x, y))

and

I4(h, l) = (−1)^{n}
Z

(IR^{n})^{2}

h(−x+y)Ψ_{l,n−l}(−X,−Y;d(x, y)).

Obviously, the measures Ψl,n−l(±X,±Y;·) are non-negative. Therefore we can deduce the integrability of the mapt7→χ(∂X∩(∂Y +t)) as soon as|Ij(h, l)|<∞has been

Since mixed volumes are invariant with respect to simultaneous applications of volume preserving affine maps, we are finally led to

I_{1}(h_{0}, l) +. . .+I_{4}(h_{0}, l)

This proves the assertion of integrability. Moreover, combining (3.4), (3.5) and (3.6), we also conclude the proof of the translative integral formula for the intersections of boundaries of parallel sets.

The argument for the intersection of boundary and body is analogous, actually it is simpler since we merely have to apply the decomposition

N(∂X)×N(Y) = (N(X)×N(Y))∪({(x,−m) : (x, m)∈N(X)} ×N(Y)) The last summand on the right-hand side can be treated as the remaining summands if we use the representation

for Borel setsA⊆IR^{n}; confer [36] or [19]. The proof can then be completed as before
by either using that (for this particular normalization of the curvature measures)

C0(∂X,(IR^{n})^{2}) =χ(S^{n−1}) = 1−(−1)^{n}

or by recalling the well-known fact thatC0(X,(IR^{n})^{2}) = 1. 2
It remains to establish the general case of Theorems 1.1 and 1.2. This is accomplished
by the following

P r o o f . Fix two convex bodies K, L ⊂ IR^{n}. First, we consider the case of the
intersection of boundaries. Choose a decreasing sequenceri∈(0,1] satisfyingri →0
fori→ ∞andr1= 1. By T^{∗}we denote the set of allt∈IR^{n} for which there is some
r∈ {ri|i∈IN} ∪ {0}such that∂Krand∂Lr+tdo not intersect almost transversally.

Lemma 3.1 implies thatH^{n}(T^{∗}) = 0. Forr≥0 we set
υr(t) :=χ(∂Kr∩(∂Lr+t)).

Lett∈IR^{n}−T^{∗} be fixed for the moment. We distinguish several cases.

Case 1: K^{◦}∩(L^{◦}+t)6=∅.

Case 1.1: ∂K∩(∂L+t)6=∅. Then∂K and∂L+t must intersect almost
transver-sally, sincet /∈T^{∗}. Thusυ_{0}(t) =υ_{r}_{i}(t), for alli∈IN, by Theorem 1.3.

Case 1.2: ∂K∩(∂L+t) =∅. It is easy to check that thenK⊆L^{◦}+torL+t⊆K^{◦}.
Hence, Kr ⊆ L^{◦}_{r} +t for all r ≥ 0 or Lr+t ⊆ K_{r}^{◦} for all r ≥ 0, and therefore
υ0(t) =υr_{i}(t) = 0 for alli∈IN.

Case 2: K^{◦}∩(L^{◦} +t) = ∅. Then ∂K ∩(∂L+t) = ∅, since t /∈ T^{∗}. Hence,
K∩(L+t) =∅. Note thatKr_{1} = (Kr)r_{1}−r and Lr_{1} = (Lr)r_{1}−r, forr∈[0, r1]. Let
r∈ {ri:i∈IN}. Using again the assumption thatt /∈T^{∗}and Theorem 1.3, we obtain

υr(t) =

υ1(t) ifKr∩(Lr+t)6=∅,

0 otherwise,

and

i→∞lim υr_{i}(t) = 0 =υ0(t).

In any case, fort /∈T^{∗} we obtain that

|υr(t)| ≤ |υ1(t)|, forr∈ {ri:i∈IN} ∪ {0}, and

i→∞lim υ_{r}_{i}(t) =υ_{0}(t).

Lemma 3.2 shows that υ1(·) is integrable with respect to H^{n}. By the dominated
convergence theorem, we thus conclude that

i→∞lim Z

IR^{n}

χ(∂Kr_{i}∩(∂Lr_{i}+t))H^{n}(dt) =
Z

IR^{n}

χ(∂K∩(∂L+t))H^{n}(dt).

An application of Lemma 3.2 then completes the proof in the case considered, since the mixed volumes are continuous in each component with respect to the Hausdorff metric.

The case of the intersection of boundaries and bodies can be treated similarly.
In-stead of Theorem 1.3 one now has to use Theorem 1.4. Moreover, Case 1.2 has to be
split into two subcases due to the asymmetry of the situation. IfKr⊆L^{◦}_{r}+t for all
r≥0, then

χ(∂Kr∩(Lr+t)) =χ(S^{n−1}) =χ(∂K1∩(L1+t))

for allr≥0. IfLr+t⊆K_{r}^{◦} for allr≥0, then χ(∂Kr∩(Lr+t)) =χ(∅) = 0 for all

r≥0. 2

Acknowledgements

We wish to thank Professor Peter Mani-Levitska for inviting us to Bern and for several interesting discussions related to the present subject.

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Daniel Hug, Reiner Sch¨atzle Mathematisches Institut Albert-Ludwigs-Universit¨at Eckerstraße 1

D-79104 Freiburg Germany