Almost Transversal Intersections of Convex Surfaces and Translative Integral Formulae
By Daniel Hug∗) of Freiburg, Peter Mani–Levitskaof Bern andReiner Sch¨atzle of Bonn
(Received May 14, 2001; revised version April 22, 2002; accepted April 25, 2002)
Abstract. This work consists of three parts, the first and second of which are concerned with generalized forms of two conjectures byWm. J. Firey(1978). LetK, L⊂Rnbe compact convex sets which have common interior points and intersect almost transversally. LetKr, Lr,r≥0, denote the outer parallel bodies ofK, Lat distancer and setSr :=∂Kr∩∂Lr,Hr:=∂Kr∩Lr. It has been conjectured byFireythat the Euler characteristic of these sets is independent ofr >0. More generally, it has been shown in [10] thatSrandS0as well asHrandH0are homotopy equivalent for allr≥0. In the present work, we prove that, for allr≥0,SrandS0as well asHrandH0are in fact bi–lipschitz homeomorphic lipschitz submanifolds ofRn. In the second part, we establish translative integral geometric formulae involving such intersections for arbitrary pairs of compact convex sets.
Formulae of this type have recently been proved in [10] for compact convex sets with interior points, while very special cases of these have been conjectured byFirey. Finally, the last part is devoted to a theoretical study of general convex surfaces in stochastic geometry.
1. Introduction
In 1978, Wm. J. Fireycontributed two conjectures to a collection of open prob- lems in geometric convexity (see [9]). The setting for these conjectures is the Euclidean space Rn, n ≥1, with scalar product ·,· and norm | · |. Let K, L⊂ Rn be com- pact convex sets with nonempty interiors and boundaries ∂K, ∂L. For r ≥ 0 let Kr, Lr denote the set of points of Rn whose respective distance from K, L is r at the most. ThenFireyconjectured that theEuler characteristicsχ(∂Kr∩∂gLr) and χ(∂Kr∩gLr), defined as in singular homology theory, are independent ofr > 0, at least for µ almost all g ∈ G(n), where µ is a Haar measure on the group G(n) of proper rigid motions of Rn and n≥2. Fireyalso suggested to use such a result in
2000Mathematics Subject Classification. Primary: 52A20, 52A22, 60D05; Secondary: 49J52, 53C65, 57R50, 57Q65, 53A07, 53C40.
Keywords and phrases. Transversal intersection, transversal field, convex surface, lipschitz man- ifold, nonsmooth analysis, Euler characteristic, integral geometry, stochastic geometry.
∗)Corresponding author/daniel.hug@math.uni-freiburg.de
c 2002 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim, 0025-584X/02/246-24712-0121 $ 17.50+.50/0
order to obtain a conjectured extension of the principal kinematic formulaof integral geometry to generalconvex surfaces; see [10] for further background information.
Both conjectures have recently been established in generalized forms in [10]. In order to state precise results, we need some definitions. Let K, L⊂Rn be compact, convex sets. We say that K and L intersect almost transversallyif K andL have common interior points and if, for all p ∈ ∂K ∩∂L, the normal cone of K at p, N(K, p), intersects N(L, p) only in the zero vector (see [15] for further definitions). Let Hn denote n–dimensional Hausdorff measure in Rn. It is an important fact that K and L+tintersect almost transversally, forHn almost allt∈Rnsuch thatK∩(L+t)=∅; see [10], Lemma 3.1. Now we set Sr :=∂Kr∩∂Lr and Hr := ∂Kr∩Lr for r ≥0.
Whereas Firey’s first conjecture concerned the Euler characteristics ofSr and Hr, for r >0, it was proved in [10], Theorems 1.3 and 1.4, that thehomotopy types, and therefore the Euler characteristics, of these intersections are independent ofr≥0 ifK and Lintersect almost transversally. The crucial points here are that the case r= 0 is included and the almost transversal intersection of two convex surfaces is a natural and convenient geometric condition. Indeed, these points are involved in an essential way in the derivation of translative integral geometric formulae.
The topological results established in [10], and the additional information available from the corresponding proofs (described subsequently), suggest some further study which is carried out in the first part of the present contribution. In the following, we always assume that K, L⊂ Rn are compact, convex sets which intersect almost transversally. Under this assumption, it is shown in [10] that Sr, for r ≥ 0, is an (n−2)–dimensional compactlipschitz submanifoldofRnwithout boundary,Sris even aC1submanifold forr >0, andSris homeomorphic toSsvia abi–lipschitz mapfor all r, s >0. Moreover, it is proved thatHr, forr≥0, is an (n−1)–dimensional compact lipschitz submanifold ofRnwith∂Hr=Sr,Hris aC1submanifold forr >0, andHr is homeomorphic to Hsvia a bi–lipschitz map for allr, s >0. Therefore the following questions naturally arise.
Are Sr and Ss diffeomorphic of class C1 for all r, s >0?
Is Sr homeomorphic toS0 via a bi–lipschitz map for all r >0?
These questions and their analogues for intersections of boundaries and bodies are solved affirmatively in Section 2, where we prove the following two main theorems (see also Corollaries 2.12 and 2.13). Here we write X ∼= Y in order to indicate that the topological spaces X andY are homeomorphic.
Theorem 1.1. Let K, L⊂Rn, n ≥2, be two compact convex sets with common interior points which intersect almost transversally. Define the intersections
Sr := ∂Kr∩∂Lr, r ≥ 0.
Then all Sr are bi–lipschitz homeomorphic to each other; in particular, Sr ∼= Ss for all r , s ≥ 0.
For the intersections of boundaries and bodies, we shall prove:
Theorem 1.2. Let K, L⊂Rn, n ≥2, be two compact convex sets with common interior points which intersect almost transversally. Define the intersections
Hr := ∂Kr∩Lr, r ≥ 0.
Then all Hr are bi–lipschitz homeomorphic to each other; in particular, Hr ∼= Hs for all r , s ≥ 0.
Compared to the results in [10], the main novelty here is the inclusion of the case r = 0 and thus a positive answer to the second question raised before. In addition, our method of proof will lead to an affirmative resolution of the preceding question concerning the diffeomorphy of the setsSr (resp.Hr), forr >0.
A new element in our approach to these results is the construction of a strongly transversal pair of lipschitz vector fieldsforS0(a definition is given in Section 2). The related notion of a field of transverse planes to a lipschitz manifold has been studied by J. H. C. Whitehead in [25]. Such a pair of lipschitz vector fields is used to obtain a lipschitz parametrization of atubular neighbourhoodofS0. As two additional transverse parameters we finally introduce the signed distance functions of ∂K and
∂L. To accomplish the required transition between different descriptions of a tubular neighbourhood ofS0we use tools fromnonsmooth analysissuch asinverseandimplicit function theoremsfor lipschitz maps.
Transverse fields appeared in the context of early smoothing theory. In particular, re- fining and extending previous work byWhitney[26] andCairns[2],J. H. C. White- headshowed that the existence of a transverse field for some topological manifoldM inRnimplies thatM has a smooth atlas, although not necessarily as a submanifold of Rn. It follows, from our results, that every almost transversal intersection of two gen- eral convex surfaces has a smooth atlas. We do not know whether a similar statement remains true if the number of convex surfaces increases to three, or more.
A major motivation to study topological properties of intersections of convex sur- faces comes from an attempt to extend global kinematic integral geometric formulae involving the Euler characteristic to arbitrary convex surfaces. To be more specific, forn≥2 Fireyconjectured the formula
G(n)
χ(∂K∩gL)µ(dg) = 1 κn
n−1 k=0
n k
1−(−1)n−k
Wn−k(K)Wk(L), together with its counterpart for intersections of two boundaries, whereκndenotes the volume of then–dimensional Euclidean unit ballBn and the functionalsW0, . . . , Wn are the classicalquermassintegrals(Minkowski functionals). The Minkowski function- als are very special examples of mixed volumes (see [15], and [10] for a short exposi- tion). The latter are in fact involved in the following more generaltranslative integral geometric formula,
Rn
χ(∂K∩(L+t))Hn(dt)
= n
i=0
n i
V(K[n−i],−L[i]) + (−1)i−1V(K[n−i], L[i]) , (1.1)
which has been established in [10] for compact, convex setsK, L⊂Rnwith nonempty interiors and n ≥ 2. There also the following integral formula has been obtained, under the same assumptions, for the intersections of two convex surfaces:
Rn
χ(∂K∩(∂L+t))Hn(dt)
= (1 + (−1)n) n i=0
n i
V(K[i],−L[n−i]) + (−1)i−1V(K[i], L[n−i]) . (1.2)
Equation (1.2) can be deduced from equation (1.1), once the former has been estab- lished and one knows thatH0andS0are lipschitz submanifolds ofRnwith∂H0=S0
(compare [6], Corollary 8.8). In [10], the proof of both results is based on a com- bination of analytic, topological and integral geometric arguments. The topological results require the restriction to sets with nonempty interiors. In this paper, we extend both equations to arbitrary compact, convex sets by means of some additional integral geometric considerations. In retrospect, it would have been possible to derive these extensions in [10], on the basis of the results and tools provided there.
Henceforth, we write∂K for therelative boundaryof a compact convex setK⊂Rn. Theorem 1.3. LetK, L⊂Rn,n≥1, be compact convex sets, and set k:= dimK, l := dimL. Then the map t→χ(∂K ∩(∂L+t)) is integrable with respect toHn on Rn and
Rn
χ(∂K∩(∂L+t))Hn(dt)
=
1 + (−1)k+l−nn
i=0
n i
V(K[i],−L[n−i]) + (−1)n−l+i−1V(K[i], L[n−i]) . (1.3)
Theorem 1.4. LetK, L⊂Rn,n≥1, be compact convex sets, and set k:= dimK.
Then the mapt→χ(∂K∩(L+t))is integrable with respect toHn on Rn and
Rn
χ(∂K∩(L+t))Hn(dt)
= n i=0
n i
V(K[i],−L[n−i]) + (−1)k−i−1V(K[i], L[n−i]) . (1.4)
The range of the indices in (1.1) and (1.2) is extended formally, in comparison with formulae (1.5) and (1.4) in [10], but the additional summands all vanish. In a similar way, the summation in Theorem 1.4 can be restricted to i ∈ {0, . . . , k−1}. The proof of Theorem 1.4 will be given in Section 3, Theorem 1.3 can be established by similar arguments. However, in the case of lower–dimensional compact, convex sets, it does not seem to be possible to deduce Theorem 1.3 directly from Theorem 1.4. In Section 3, we shall also treatiterated translative integral formulaeand an extension of Theorem 1.4 to elements Lof theconvex ring. Such additional results are important prerequisites for applications instochastic geometry.
The final section is designed to provide a theoretical study of convex surfaces in stochastic geometry. Our intention is to demonstrate in an exemplary way how the
integral geometric results obtained in Section 3 can be used to extend known results and methods to admit the treatment of convex surfaces as basic particles of point processes or as test sets. This is not completely straightforward, since the Euler characteristic is not alocally boundedfunctional (cf. [24], p. 332, or [18], p. 184) in the present setting.
2. Homeomorphy via transversal fields
Throughout this section we consider two compact convex sets K, L ⊂ Rn with common interior points, that is
(2.1) intK∩intL = ∅,
which intersect almost transversally, that is which satisfy N(K, p)∩N(L, p) = {0} (2.2)
and
N(K, p)∩(−N(L, p)) = {0} (2.3)
for allp∈∂K∩∂L. Clearly, condition (2.3) is implied by (2.1).
We put M := ∂K ∩∂L and know from [10] that M is an (n−2)–dimensional lipschitz manifold without boundary. (The reader should be warned that the symbol M appears with a different meaning in [10].)
Subsequently, Bn(a, r) denotes the Euclidean unit ball of radiusr ≥ 0 centred at a∈Rn, andSn−1:=∂Bn. Further notation will be introduced when necessary.
2.1. Construction of strongly transversal fields
We define the open convex tangent cone (inner cone) ofK atp∈K by I(K, p) := {λ(y−p)|y∈intK, λ >0}.
Roughly speaking, normal cones and tangent cones are connected by duality of convex cones. This connection will be used in the proof of the following lemma.
Lemma 2.1. Let K, L⊂Rn be two compact convex sets with nonempty interiors, and letp∈∂K∩∂L. Then (2.2)is satisfied if and only if
I(K, p)∩(−I(L, p)) = ∅, and (2.3)is satisfied if and only if
I(K, p)∩I(L, p) = ∅.
P r o o f . We start with some preliminary remarks. Let us define the support cone S(K, p) ofK atpas in [15] by setting
S(K, p) := cl{λ(y−p)|y∈K, λ >0},
where “cl” designates the formation of the topological closure. Obviously, we have I(K, p) ⊆S(K, p) and clI(K, p) = S(K, p). By Theorem 1.1.14 in [15] we thus see that I(K, p) = intS(K, p).
It is sufficient to prove the first assertion. Obviously, condition (2.2) is violated if and only if there is some u∈Rn\ {0}such that
{λu|λ≥0} ⊆N(K, p)∩N(L, p) or equivalently
(N(K, p)∩N(L, p))∗⊆ {z∈Rn| z, u ≤0}, (2.4)
where “∗” denotes the formation of the dual cone; see [15, p. 34]. By Theorem 1.6.3 and equation (2.2.1) in [15], it follows that
(N(K, p)∩N(L, p))∗ = S(K, p) +S(L, p), and hence condition (2.4) is satisfied if and only if
(2.5) S(K, p) +S(L, p)⊆ {z∈Rn| z, u ≤0}
for some u∈Rn\ {0}. It can easily be inferred from the separation Theorem 1.3.8 in [15] that condition (2.5) is fulfilled if and only if I(K, p)∩(−I(L, p)) = ∅, which
completes the proof. 2
In [25],Whitehead defined the notion of a field of transverse planes to a manifold.
Specifying a basis for each of the transverse planes, we find the following definition useful.
Definition 2.2. A pair of continuous mappingsx, y:M →Rn such that, for each p∈M,
(2.6) x(p)∈I(K, p)∩I(L, p) and y(p)∈I(K, p)∩(−I(L, p))
is called a strongly transversal pair of vector fields (forM). If x andy are lipschitz mappings, then the pairx, yis said to be astrongly transversal pair of lipschitz vector fields(forM).
The existence of such vector fields is shown in the next proposition.
Proposition 2.3. There exists a strongly transversal pair of lipschitz vector fields.
Moreover, let x0∈intK∩intL be arbitrarily fixed. Then xcan be chosen such that
(2.7) x(p) = x0−p , p∈M .
P r o o f . Clearly, a map xdefined as in (2.7) is lipschitz and x(p)∈I(K, p)∩I(L, p) for allp∈M.
Next we construct the mapy. We consider anyp0 ∈M. SinceK andL intersect almost transversally and by Lemma 2.1, we can select
y0∈I(K, p0)∩(−I(L, p0)) = ∅.
By definition we know that there existλ, µ >0 such that p0+λy0∈intK and p0−µy0∈intL .
As intK and intLare open und nonempty, there exists an open neighbourhoodU(p0) of p0 inRn such that
p+λy0∈intK and p−µy0∈intL , for allp∈U(p0)∩M. Therefore
y0 = λ−1(p+λy0−p)∈I(K, p),
−y0 = µ−1(p−µy0−p)∈I(L, p), and hence
y(p0) := y0∈I(K, p)∩(−I(L, p)), (2.8)
for allp∈U(p0)∩M. AsM is compact, we may select a finite cover
M =
N i=1
(U(pi)∩M)
and a subordinated partition of unityϕi∈C0∞(U(pi)),i∈{1, . . . , N}, with 0≤ϕi≤1 and
N i=1
ϕi = 1 on M . Then we define
y(p) :=
N i=1
ϕi(p)y(pi), p∈M .
Clearly, y :M →Rn is lipschitz. Now letp∈M be fixed for the moment. Then we have indices 1≤i1< . . . < il≤N such that
p∈U(pi), i∈ {i1, . . . , il}, and
p /∈U(pi), i /∈ {i1, . . . , il}. Hence (2.8) implies that
y pij
∈I(K, p)∩(−I(L, p)), j∈ {1, . . . , l}, and
l j=1
ϕij(p) = 1. This yields
y(p) = l j=1
ϕij(p)y pij
∈I(K, p)∩(−I(L, p)),
since the open tangent cones are convex. This proves the assertion, since p∈M was
arbitrarily chosen. 2
Remark 2.4. The proof of Proposition 2.3 is similar to the proof of Michael’s selection theorem (see [12], Theorem 3.2, or Theorem 1.6 in [1]). Instead of trying to apply such an abstract result, we decided to proceed in a more direct (and more elementary) way, which also yields additional information. In fact, if K andLare of class C1, then M is of class C1 and so are the lipschitz vector fields constructed in Proposition 2.3.
2.2. Lipschitz parametrization of a tubular neighbourhood
The strongly transversal pair of lipschitz vector fields of the previous subsection will be used to parametrise a tubular neighbourhood of M. We recall that M has codimension 2. As additional two parameters we can choose the signed distance func- tions of ∂K and∂L(see Proposition 2.9), where the signed distance function d∂Z of a nonempty set Z⊂Rn is defined by
d∂Z(x) := d(Z, x)−d(Rn\Z, x), x∈Rn. We start with a preparatory definition.
Definition 2.5. Letx, y:M →Rnbe a strongly transversal pair of lipschitz vector fields forM. Then lipschitz maps Φ, F are defined by
Φ : M×R2 −→ Rn, Φ(p, λ, µ) := p+λx(p) +µy(p), (2.9)
and
F : M ×R2 −→ R2, F(p, λ, µ) := (d∂K, d∂L)(Φ(p, λ, µ)). (2.10)
The desired parametrization will be constructed by using the inverse and implicit function theorems for lipschitz maps (see [4], Section 7) and the maps Φ andF. To this end, we have to calculate several generalized Jacobians in the next three lemmas.
We first turn to Φ which gives a parametrization of an open neighbourhood in terms of the transverse parameters λ, µ.
Lemma 2.6. For each p0∈M there exists an open neighbourhood V ⊆M×R2 of (p0,0)∈M×R2such thatΦmapsV homeomorphically and bi–lipschitz onto an open neighbourhood of p0∈Rn.
P r o o f . Let p0 ∈ M be fixed. Recall that M is an (n−2)–dimensional lipschitz manifold without boundary (see [10]).
First, we shall choose a simple bi–lipschitz chart onto an open neighbourhood of p0 ∈ M. Second, we compute the generalized Jacobian of Φ in (p0,0), according to [4], and we show that this Jacobian has full rank. Finally, the lemma is deduced by the application of an inverse function theorem for lipschitz maps; see [4], Theorem 7.1.1.
Following the proof of Proposition 2.2 in [10], we obtain more precise information about the form of a particular chart. To simplify the notation, we assume that (2.11) x(p0) = γen and y(p0) = αe1+βen,
where γ >0,α <0 and (e1, . . . , en) is an orthonormal basis ofRn.
By the definition of the inner cone, there is some λ >0 such that p0+λx(p0) =:
x0∈intK∩intL. Thus we see that there areδ >0 ,x0= (y0, t0) and convex lipschitz maps
f, g : U(y0, δ) −→ R,
where U(y0, δ)⊂Rn−1denotes an open ball of radiusδcentred aty0, such that
∂K∩(U(y0, δ)×(−∞, t0)) = graph(f|U(y0, δ)) and
∂L∩(U(y0, δ)×(−∞, t0)) = graph(g|U(y0, δ)).
Clearly, for vectors v ∈ ∂f(y0) andw ∈∂g(y0) in the generalized subgradient (sub- differential) off, g, respectively, we get
(v,−1)∈N(K, p0) and (w,−1)∈N(L, p0). Since y(p0)∈I(K, p0)∩(−I(L, p0)), we see that
−(v,−1), y(p0) ≥ µ0|(v,−1)| ≥ µ0 > 0 and
(w,−1), y(p0) ≥ µ0|(w,−1)| ≥ µ0 > 0
for someµ0>0 independent of the particular choice ofv, w. Then (2.11) implies that αv, e1 ≤ −µ0+β ≤ αw, e1 −2µ0.
In particular, as µ0>0> α, there isµ >0 such that
v−w, e1 ≥ µ for v∈∂f(y0), w∈∂g(y0). Putting
Λ(y) := f(y)−g(y),
we get as in [10], Proposition 2.2, (see also [4], Theorem 2.5.1) that ξ, e1 ≥ µ for ξ∈∂Λ(y0) ;
in particularξ= 0. Here the generalized gradient∂Λ(y0) of the lipschitz map Λ aty0 is defined as in [4], p. 27.
Now applying the implicit function theorem (see [4], Corollary 7.1.3), we get as in [10], Proposition 2.2, that locally
[f =g] = [Λ = 0] = {(ζ(z), z)}
for some lipschitz function ζ : U(z0) ⊆ Rn−2 → R which is defined on an open neighbourhood U(z0) of somez0∈Rn−2. Putting
h(z) := (ζ(z), z, f(ζ(z), z)) = (ζ(z), z, g(ζ(z), z)), we obtain the bi–lipschitz chart
h : U(z0)⊆Rn−2 −→ U(p0)⊆M , h(z0) = p0,
we were looking for. Note that here and in the following we do not strictly distinguish between row and column vectors. We define
Φ(z, λ, µ) := Φ(h(z), λ, µ) =ˆ h(z) +λx(h(z)) +µy(h(z)),
and we have to prove that ˆΦ maps an open neighbourhood of (z0,0,0) onto an open neighbourhood ofp0∈Rn. To this end, we want to invoke the inverse function theorem [4], Theorem 7.1.1. We have to establish that each member
A∈∂Φ(zˆ 0,0,0)
of the generalized Jacobian of ˆΦ has full rank. Actually, to avoid unnecessary compli- cations on sets of measure zero, we consider the smaller set
(2.12)
∂SΦ(zˆ 0,0,0) := conv
i→∞lim DΦ(zˆ i, λi, µi)|(zi, λi, µi)→(z0,0,0), h, x◦h, y◦hare differentiable atzi
,
where “conv” denotes the formation of the convex hull; see [4], Proposition 2.6.4. We recall that lipschitz functions are differentiable almost everywhere by Rademacher’s theorem.
We will show that
(2.13) Av = 0 for A∈∂SΦ(zˆ 0,0,0), v∈Rn\ {0}. We consider (zi, λi, µi) as in (2.12) and calculate
B ←− DΦ(zˆ i, λi, µi) =
∇ζ(zi)
In−2 x(h(zi)) y(h(zi)) (∇zf(ζ(·),·))(zi)
+λi(D(x◦h)(zi),0,0) +µi (D(y◦h)(zi),0,0). Since ∇ζ,∇zf(ζ(·),·), D(x◦h), D(y◦h) are bounded andλi, µi→0, we see that
B =
ξ 0 α In−2 0 0
η γ β
for someξ, η∈Rn−2.
Since this form of B is closed under taking convex combinations, we infer that all A ∈∂SΦ(zˆ 0,0,0) have the form above. In particularA is nonsingular, as α, γ = 0,
which implies (2.13), and the lemma is proved. 2
To switch from the transverse parametersλ, µto the signed distance functions, we calculate the generalized derivative of the signed distance functions. Although the following lemma may be known, we shall give the proof for the reader’s convenience.
Lemma 2.7. Let p0∈∂K and c∈I(K, p0). Then (2.14) ∂d∂K(p0), c ⊆(−∞,0).
P r o o f . Assume that (2.14) is not true. Then by Theorem 2.5.1 in [4], we can find a sequence zj,j ∈N, such thatzj ∈∂K, zj →p0 as j→ ∞,d∂K is differentiable at zj, and
(2.15) lim
j→∞
∇d∂K zj
, c ≥ 0.
By Proposition 2.5.4 in [4], for eachj∈Nthere is somepj∈∂K satisfyingzj−pj= d
∂K, zj
>0 and
vj := ∇d∂K zj
= zj−pj
zj−pj if zj ∈K , vj := ∇d∂K(zj) = − zj−pj
zj−pj if zj ∈K . Now ifzj ∈K, thenK⊆
z∈Rn|
vj, z−pj
≤0
andvj∈N K, pj
. Ifzj∈intK, then we setj:=zj−pjand obtainBn
zj, j
⊆Kwithpj ∈∂Bn zj, j
. Therefore vj = pj−zj
pj−zj ∈N Bn
zj, j , pj
= N(K, pj).
Taking a subsequence, we obtainvj →v∈Sn−1. As clearlyzj−pj≤zj−p0→0, and hence pj → p0, we deduce that v ∈ N(K, p0)∩Sn−1. Since c ∈ I(K, p0) and I(K, p0) is open, this yieldsv, c<0. On the other hand, we obtain from (2.15) that
v, c ≥0, which is a contradiction. 2
With the previous lemma, we are now able to compute the generalized derivatives of the function F defined in (2.10). This allows us to switch from the transverse parameters to the signed distance functions.
Lemma 2.8. For any p0 ∈M, there exist an open neighbourhood U(p0)⊆ M of p0∈M,δ0>0, an open neighbourhoodW of p0 inRn, and a lipschitz map
Ψ : U(p0)×(−δ0, δ0)2 −→ W
that maps U(p0)×(−δ0, δ0)2homeomorphically and bi–lipschitz ontoW in such a way that
d∂K(Ψ(p, 1, 2)) = 1 and d∂L(Ψ(p, 1, 2)) = 2
for p∈U(p0)and |1|,|2|< δ0. P r o o f . We shall define
(2.16) Ψ(p, 1, 2) := Φ(p, λ(p, 1, 2), µ(p, 1, 2))
where λ, µ:U(p0)×(−δ0, δ0)2 →Rare lipschitz maps obtained by an application of the implicit function theorem for lipschitz maps (compare [4], Corollary 7.1.3) to the map F:Rn−2×R2×R2→R2defined by
F(p, 1, 2, λ, µ) := (d∂K, d∂L)◦Φ(p, λ, µ)−(1, 2). The functions λ(·),µ(·) will then satisfy
(2.17) F(p, λ(p, 1, 2), µ(p, 1, 2)) = (1, 2)
for p∈U(p0) and|1|,|2|< δ0 after choosing suitableU(p0)⊆M andδ0>0. This will establish the conclusion of the lemma.
To apply the implicit function theorem, we have to verify that the (projected) gen- eralized Jacobian (cf. [4], p. 256)
πλ,µ∂F(p 0,0,0)⊆R2,2
has full rank in the sense of [4], p. 253. In order to verify this condition on the rank, we define the lipschitz functions
ϕ(p, λ, µ) := d∂K(Φ(p, λ, µ)) = d∂K(p+λx(p) +µy(p)) and
ψ(p, λ, µ) := d∂L(Φ(p, λ, µ)) =d∂L(p+λx(p) +µy(p)).
By the definition of the generalized Jacobian (compare [4], Proposition 2.6.2), we get πλ,µ∂F(p0,0,0)⊆
∂λϕ(p0,0,0) ∂µϕ(p0,0,0)
∂λψ(p0,0,0) ∂µψ(p0,0,0)
.
Here the right–hand side is defined as the set of all matrices a b
c d
∈ R2,2 with a∈∂λϕ(p0,0,0),b∈∂µϕ(p0,0,0),c∈∂λψ(p0,0,0) andd∈∂µψ(p0,0,0). Now by the chain rule of nonsmooth analysis (see [4], Theorem 2.6.6), and by Lemma 2.7, we get
∂λϕ(p0,0,0)⊆conv (∂d∂K(p0), x(p0))⊆(−∞,0), and likewise
∂µϕ(p0,0,0)⊆conv (∂d∂K(p0), y(p0))⊆(−∞,0),
∂λψ(p0,0,0)⊆conv (∂d∂L(p0), x(p0))⊆(−∞,0),
∂µψ(p0,0,0)⊆conv (∂d∂L(p0), y(p0))⊆(0,∞), where the last inclusion is due to the fact that−y(p0)∈I(L, p0).
This indeed yields thatπλ,µ∂F(p0,0,0) has full rank. 2 Lemma 2.6 and Lemma 2.8 both yield parametrizations locally in an open neigh- bourhood of (p0,0)∈M×R2. By standard compactness and covering arguments, we
now extend these results to global parametrizations of a tubular neighbourhood ofM inRn.
Proposition 2.9. Let x, y : M → Rn be a strongly transversal pair of lipschitz vector fields forM. Then there exist an open neighbourhoodV ⊆M ×R2of M× {0} and δ >0 such that
(i) the map Φ, defined in (2.9), maps V homeomorphically and bi–lipschitz onto Uδ(M) :={x∈Rn | |d∂K(x)|,|d∂L(x)|< δ}, that isV ∼=Uδ(M)⊆Rn;
(ii)there is a lipschitz map
Ψ :M×(−δ, δ)2 −→ Uδ(M)
that maps M×(−δ, δ)2 homeomorphically and bi–lipschitz onto Uδ(M)in such a way that
(2.18) d∂K(Ψ(p, 1, 2)) = 1, d∂L(Ψ(p, 1, 2)) = 2 for p∈M and |1|,|2|< δ;
(iii)for allp∈M,
Bp : = Φ({(q, λ, µ)∈V | q=p})
= Ψ({(q, 1, 2)| q=p,|1|,|2|< δ}) (2.19)
is a two–dimensional open topological ball contained in Ep:=p+ lin{x(p), y(p)}. P r o o f . We have already seen that Φ is a local, bi–lipschitz homeomorphism in an open neighbourhood ofM×{0}. By [25], Lemma 4.1, there is an open neighbourhood V ⊆ M ×R2 of M × {0} that is mapped homeomorphically by Φ onto an open neighbourhood U ⊆Rn ofM. Ifδ >0 is small enough, thenUδ(M)⊆U and we can choose V := Φ−1(Uδ(M)).
Likewise, bi–lipschitz mappings satisfying (2.18) were obtained locally in Lemma 2.8.
SinceM is compact, we can select a finite cover of open neighbourhoodsU pj
⊆M, open neighbourhoods Wj ofpj inRn,δj >0 and bi–lipschitz mappings
Ψj : U pj
×
−δj, δj2
−→ Wj as in Lemma 2.8, which satisfy (2.18). ChoosingU
pj
andδ <minjδj small enough such that the Ψj coincide on the intersections ofU
pj
(compare (2.16) and (2.17)), we can paste together a lipschitz map Ψ : M ×(−δ, δ)2 → Rn which is a local, bi–lipschitz homeomorphism in Rn. Note that by construction Ψ(p,0,0) =p for all p∈M. Hence Lemma 4.1 in [25] can be applied again. The assertions (i) and (ii) now follow ifδ >0 is possibly reduced further.
The remaining assertion (iii) follows from (2.16) and (2.17). 2 Remark 2.10. If K and Lare parallel bodies of compact convex sets C, D, that is, K =Cr, L=Ds for some r, s >0, then the previous constructions show that Φ and Ψ can be chosen to be of classC1(compare Remark 2.4). In fact, in this case the distance functions d∂K andd∂L are differentiable in a neighbourhood of∂K and∂L, respectively.
Lemma 2.11. Let γ : [0,1]→Ep∩Uδ(M)be continuous, and assume that γ(0) ∈ Bp. Thenγ([0,1])⊆Bp.
P r o o f . Define A := {t ∈ [0,1] | γ(t) ∈ Bp}. Note that A = ∅ and that A is open, since γ is continuous and Bp is open inEp∩Uδ(M). To complete the proof, we show that A is also closed. Hence let tj ∈ A for j ∈ N and tj → t ∈ [0,1] as j → ∞. Thenγ
tj
∈Bp, for allj ∈N, and therefore there areλj, µj ∈Rsuch that γ
tj
= Φ
p, λj, µj and
p, λj, µj
∈ V. Since γ(t) ∈ Uδ(M) for all t ∈ [0,1], we obtain
p, λj, µj
= Φ−1 γ
tj
. But Φ :V →Uδ(M) is a homeomorphism, and thus Φ−1(γ(t)) = lim
j→∞Φ−1 γ
tj
= lim
j→∞
p, λj, µj
= (p, λ, µ)
for someλ, µ∈Rwith (p, λ, µ)∈V. This shows thatγ(t) = Φ(p, λ, µ)∈Bp. 2 2.3. Proof of bi–lipschitz homeomorphy
In this subsection, we prove Theorems 1.1 and 1.2 on the homeomorphy of intersec- tions of convex bodies (compact convex sets with nonempty interiors) and boundaries of convex bodies.
Proposition 2.9 already provides all tools which are needed for the proof of Theo- rem 1.1.
P r o o f of Theorem 1.1. From [10], Equation (2.7), we know that Sr ∼=Ss for r , s > 0.
Moreover, the homeomorphism was provided by a bi–lipschitz map. Therefore, it suffices to prove that S0 can be mapped onto Sr by a bi–lipschitz homeomorphism, for some (small)r >0.
We consider the lipschitz map
T : M = S0 −→ Sr, defined by
T(p) := Ψ(p, r, r),
where Ψ is from Proposition 2.9 and 0< r < δ. From (2.18), we get (d∂K, d∂L)(T(p)) = (r, r),
hence T(p)∈Sr.
On the other hand, ifx∈Sr⊆Uδ(M), whereUδ(M) is as defined in Proposition 2.9, there exists (p, 1, 2)∈M×(−δ, δ)2 such that
Ψ(p, 1, 2) = x ,
since Ψ :M×(−δ, δ)2→Uδ(M) is bijective. Sincex∈Sr and using (2.18) again, we find
(r, r) = (d∂K, d∂L)(x) = (d∂K, d∂L)(Ψ(p, 1, 2)) = (1, 2),
hence 1=2=randx=T(p).
This proves that T maps S0 ontoSr. Since T is clearly injective, as Ψ is injective, and S0and Srare compact,T is a bi–lipschitz homeomorphism as required. 2 The difficulty in the case where a body intersects with a boundary is that we have to consider also points which do not lie in the tubular neighbourhood Uδ(M) which we have parametrised in Proposition 2.9 with the signed distance functions. Clearly, we have to consider pointsxwithd∂L(x)<−δ.
Therefore we establish the existence of the homeomorphism for Theorem 1.2 in two steps. First, we apply a homeomorphism ∂K → ∂Kr induced by normal fields, and then we “stretch smoothly out” on ∂Kr using the parametrization of Proposition 2.9 in order to get that the image ofH0 is indeedHr. All these transformations will also be bi–lipschitz.
P r o o f of Theorem 1.2. From [10], Equation (2.17), we know that Hr ∼= Hs for r , s > 0.
In fact, the proof in [10] even shows that Hr and Hs can be transformed into each other by a bi–lipschitz map, for allr, s >0. Therefore, it suffices to prove thatH0can be mapped ontoHrby a bi–lipschitz homeomorphism, for some (small)r >0.
Fix x0 ∈ intK ∩intL and set x(p) := x0−p for p ∈ Rn. By Proposition 2.3 there is a strongly transversal pair (x, y) of lipschitz vector fields onM. Furthermore, Proposition 2.9 ensures the existence ofδ0 ∈(0,1) and a map Ψ :M×(−δ0, δ0)2 → Uδ0(M) which is bi–lipschitz and satisfies (2.18) and (2.19), whereδis replaced byδ0
now.
We define a map Γ :∂K×[0,∞)→Rn\intK by Γ(q, r) := q+λ(q, r)(q−x0),
whereλ(q, r)≥0 is specified by the requirement thatq+λ(q, r)(q−x0)∈∂Kr. Thus Γ|∂K×[0,1] defines a bi–lipschitz correspondence. In particular,
Γr := Γ(·, r) : ∂K −→ ∂Kr
is a bi–lipschitz map, for each r ≥ 0. Hence, for any r ≥ 0, we see that H0 and Γr(H0) ⊂ ∂Kr are homeomorphic via a bi–lipschitz map. It remains to prove that Γr(H0) and Hr = ∂Kr∩Lr can be transformed into each other by a bi–lipschitz homeomorphism, for somer >0.
LetBp, forp∈M, be defined as in Proposition 2.9. Forp∈M andq ∈Bp∩∂K we observe thatq−x0= (q−p)−x(p)∈lin{x(p), y(p)}, hence
(2.20) Γr(q)∈Ep.
By the lipschitz property of Γ, there is a positive constantc >0 such that
|Γr(p)−p| ≤ c r , for allp∈∂K andr∈[0,1].
We fix 0< r < δ0 sufficiently small such that for1=δ0/4 and for all 0≤t≤1 the following conditions are satisfied:
(i) ifp∈H0, thend∂L(Γtr(p))< 1;
(ii) if p∈∂K∩[d∂L≥0], thend∂L(Γtr(p))>−1/4;
(iii) ifp∈∂K∩[d∂L<−21], thend∂L(Γtr(p))<−1; (iv) ifp∈∂K∩[d∂L≥ −21], thend∂L(Γtr(p))>−41. From (i) we infer
Γr(H0)⊆[d∂L< 1], and (ii) implies
∂Kr∩[d∂L≤ −1/4]⊆Γr(H0). (2.21)
Subsequently, we construct a bi–lipschitz homeomorphism Ξ : Γr(H0) −→ ∂Kr∩Lr = Hr. From (2.21) we deduce that
∂Kr∩[d∂L≤ −1/4] = Γr(H0)∩[d∂L≤ −1/4]. Set
Γ1 := Γr(H0)∩[d∂L≤ −31/4] and Γ2 := Γr(H0)∩[d∂L≥ −1]. Then we define
Ξ|Γ1 := idΓ1.
It remains to define Ξ on Γ2 in a consistent way. As a first step towards such a definition, we deduce a description of Γ2 in terms of Ψ. To this end, we assert that (2.22) Γ2 = {Γr(Ψ(p,0, ))|p∈M, ∈[−21,0]} ∩[d∂L≥ −1].
In fact, since Ψ(p,0, )∈H0, for anyp∈M and ∈[−21,0], the set on the right–
hand side of (2.22) is contained in Γ2. Conversely, let y ∈Γ2. Hence d∂L(y)≥ −1 and y = Γr(x) for some x ∈ H0. By (iii) we obtain d∂L(x) ≥ −21, and thus x∈∂K∩[−21≤d∂L≤0]⊆Uδ0(M). From this we conclude thatx= Ψ(p,0, ) for somep∈M and∈[−21,0], which proves (2.22).
Next we considerγ(t) := Γtr(Ψ(p,0, )),t∈[0,1], for fixedp∈M and∈[−21,0].
Then γ(0) = Ψ(p,0, ) ∈ Bp ∩∂K and γ is continuous. In addition, since γ(t) ∈
∂Ktr, d∂L(γ(t)) < 1 and d∂L(γ(t)) > −41 (compare (i) and (iv)), we infer that γ(t) ∈ Uδ0(M). From (2.20), we see that γ(t) ∈Ep. Lemma 2.11 now implies that Γr(Ψ(p,0, ))∈Bp, for allp∈M and∈[−21,0].
We define the lipschitz mapζ:M×[−21,0]→(−δ0, δ0) by putting ζ(p, ) := d∂L(γ(1)) = d∂L(Γr(Ψ(p,0, ))). We see that ζ(p, )∈(−41, 1), since−41< d∂L(γ(1))< 1, and
Γr(Ψ(p,0, )) = Ψ(p, r, ζ(p, )).
Let p ∈ M be fixed for the moment. Since Γr(Ψ(p,0,·)) is an injective map on [−21,0], the mapζ(p,·) must be strictly monotone. By (iii) and (ii) we obtain
ζ(p,−21) = d∂L(Γr(Ψ(p,0,−21))) ≤ −1 and
ζ(p,0) = d∂L(Γr(Ψ(p,0,0))) > −1 4 . Hence ζ(p,·) is increasing and therefore
ζ(p,[−21,0]) = [ζ(p,−21), ζ(p,0)].
Defining the lipschitz maph:M →[−1/4, 1] byh(p) := ζ(p,0), we finally obtain from (2.22) that
(2.23) Γ2 = {Ψ(p, r, )|p∈M, ∈[−1, h(p)]}. Next we define a bi–lipschitz map
α(p,·) : [−1, h(p)] −→ [−1, r]
which satisfies
α(p,·)|[−1,−1/2] = id[−1,−1/2]
and
α(p, t) := 1+ 2r
1+ 2h(p)t+1(r−h(p))
1+ 2h(p) , t∈[−1/2, h(p)]. Thus, in particular
α(p,−1/2) = −1/2 and α(p, h(p)) = r .
Furthermore, we set G(p, r, t) := (p, r, α(p, t)) ifp∈M andt∈[−1, h(p)].
Finally, we define
Ξ|Γ2 := Ψ◦G◦Ψ−1|Γ2.
By definition, Ψ◦G◦Ψ−1 is the identity map on Γr(H0)∩[−1 ≤d∂L ≤ −1/2], and thus Ξ is well–defined. It is also easy to see that Ξ is injective and bi–lipschitz.
Clearly, im
Ξ|Γ1
⊆Hr and im Ξ|Γ2
= {Ψ(p, r, )|p∈M, ∈[−1, r]} ⊆Hr. On the other hand,
im Ξ|Γ1
⊇∂Kr∩[d∂L≤ −31/4]
and for any y ∈ ∂Kr∩[−1 ≤d∂L ≤ r] ⊆Uδ0(M) there is some p∈ M and some ∈[−1, r] such thaty= Ψ(p, r, ); thus
im Ξ|Γ2
⊇∂Kr∩[−1≤d∂L≤r].
This shows that Ξ maps onto Hr. 2
