REND. SEM. MAT. UNIVERS. POLITECN. TORINO

Vol. 44*, 2 (1986)

Rolf Schneider

CURVATURE MEASURES AND INTEGRAL GEOMETRY OF CONVEX BODIES, II

Summary. Integral-geometric formulae for rotational means of Minkowski sums or projec­tions of convex bodies involving curvature measures on sets of boundary points or normal directions are extended to generalized curvature measures, which are de­fined on sets of support elements. Also the methods of proof contain some new aspects.

1. Introduction

For a convex body K in w-dimensional Euclidean space En, let Sj(K, \) denote its surface area measure of order i or i-th curvature measure of the second kind, and let Cj(K, •) be its i-th Federer curvature measure or cur­vature measure of the first kind (see, e.g., [7] or the survey article [10]). In [6] the integral-geometric formula

(1.1) f Si(K + pK', co H p^)dv(p) = J Q (lk)Sk(K,u)Si-k(K

,t wf)

S0n

(i = 0,... ,n - 1) was proved. Here K and K' are convex bodies, CJ and a / are Borel sets on the unit sphere, and S0n is the rotation group of En with Haar measure v (suitably normalized, see §2). Using a result on curvature measures proved in [8], Weil [16] showed that

Classificazioneper soggetto AMS (MOS, 1980): 52A22, 53C65, 60D05

264

(1.2) f Ci(K + pK', p + pf$')dv(p) = ^Ck)Ck(K^)Ci.k{K\n

for i G {0,... ,w - 1} and Borel sets j3 C dK, ft C bK*. It is the aim of the present paper to prove a common generalization of (1.1) and (1.2); see (3.3) below. This holds for the generalized curvature measures on sets of support elements, which were introduced in [9] and further studied in [11]. Such a generalization is not only of formal interest; in fact, one of its consequences is needed for certain applications in stochastic geometry which will be describ­ed elsewhere.

Our proof of the generalization uses formula (1.1). The proof given in [6] for (1.1) is rather indirect, relying as it does on a characterization theorem for the surface area measures proved earlier in [5]. Formula (1.1) is also deduced by Weil [16], but in an equally indirect way, using results from Weil [15] which in turn make use of work of Firey [1]. An application of (1.1) appears in Firey [2]. It might be of some interest to have a short and direct proof of (1.1). Such a proof is presented in § 4.

In § 5 we prove a projection formula for generalized curvature meas­ures. It extends known formulae for curvature measures of first and second kind and is obtained as a consequence of the main result of § 3.

2. Preliminaries

En is the w-dimensional Euclidean vector space with scalar product <•, *> and norm II • II, Sn~l = {x GEn: \\x\\ = 1} is its unit sphere, and 2 = £ w x 5 B _ 1 is the product space. For a topological space X> the a-al-gebra of Borel subsets of X is denoted by $ (X). By X* we denote the Lebesgue measure on &-dimensional affine subspaces of En and by ak the spherical Lebesgue measure on ^-dimensional great spheres of 571"1. The volume of the &-dimensional unit ball is given by K^ = 7r*/2/r(l +k/2). Let S0„ be the rotation group of En with Haar measure v, normalized so that v(S0n) = on~1(Sn~1) = nicn. By Gn we denote the group of proper rigid motions of En. For t^En and pGS0n let Ttp be the motion defined by Tt nX = px + t for x GEn. The image measure of \n *v under the map (t, p) I—• Tt p from En x S0n onto Gn is the Haar measure p. of Gn, suitably normalized.. For functions on S0n and Gn and for subsets, measurability refers to the completions of the measure spaces (S0n, $&(S0W), v)

265

and (G„, JB(G„), JU), respectively. For (# ,w)E2 and g = Tt p€Gn we write g(x,u) = (gx, pu).

K? denotes the space of convex bodies (non-empty, compact, convex subsets) in En, equipped with the Hausdorff metric. Let KGIC1 be given. For xGEn\K, we denote by p(K,x) the point in K nearest to x, and we write r(K,x) := \\x-p(K,x)\\ and u(K,x) := (x-p(K,x))/r(K,x). Each pair (p(K,x), u(K,x)) is a support element of K, that is, it consists of a boundary point of K and an exterior unit normal vector to K at this point; vice versa, each support element of K is obtained in this way. The set of all support elements of K is denoted by X(K).

For arbitrary sets A, A' C En, their vector or Minkowski sum is defined by A+A' = {a+a': aGA, a'EA'}. Let K,K'eiC be convex bodies. If (x,u)e2(K) and ( * » E 2 ( / 0 , then (x+x\ u)eX(K + K'). Vice versa, let (x,u) E 2(K + K'). The point xEK + K' is of the form x = — y+y' with y€K, y'GK', and necessarily (y,u)GX(K) and (xf, u) E GS(K'). In general, y and y* are not uniquely determined. If x = z4-z ' with (z ,«)G2 (K), (2', « ) G 2 (if') is a different representation, then 2 —y -= y'~z\ hence K and K' contain parallel segments lying in supporting hyperplanes with the same normal vector u. In this case we say that K and K' are in special relative position.

We define an operation for sets of support elements which is adapted to Minkowski addition of convex bodies. For 17,17' C 2 , let

T? *T?' := {(x+x\ u) G £ :(#,w)Er?, (#', « ) E T ? ' } .

In particular, for |3, j3' C £* and cj,a;' C Sw_1 we have

• t fxw)*03'xco ') = tf + |3 ' )x(conco ' ) .

If K,K'eiC are convex bodies and i ) C 2 (K), T?' C 2 (#'), then T? * T?' C

Finally, we repeat the definition of the curvature measures. For K E K", f?e?B(£), and e > 0 let

M€(K,'0):={xeEn:O<r(K,x)<e and (p(K,x), u(K,x)) ET?} .

Then Af€(/C,77) is a Borel set, and for its Lebesgue measure one has a poly­nomial expansion

266

(2.1) X"(M€(K,r))) = |-"So1e' ' - ' (^)0 )(if (r ,) ,

which defines positive measures ®0(K,'),..., @„-i(/C, •) on SMS). These are.the generalized curvature measures of K, as introduced in [9]. The cur­vature measures of first and second kind are obtained by specialization:

(2.2) d(K,e) = Bi(K, fi x S"-1) , |3 G jg(JE») ,

(2.3) S,(^,o;) = e , ( / i r ,Fx6 ; ) , o ; G P ( 5 w - 1 ) .

3. A generalized integral formula for Minkowski sums

Before we can state our main result, we need the following lemma.

(3.1) Lemma. For K.K'eiC and Borel sets r?CD(K), r?'CS(JK"), the set r\*pr\' is a Borel set for v-almostall pG50„.

Proof Let K, K', 77,r?' be given. By # we denote the set of all rotations pG 50w for which K and pK' are not in special relative position. Let p G R be given. For (y,w) G 2 (# + pK') we have 3/ = x 4- p# ' with suitable (x,u)G2(K) and (p^ ' , « )E2(pX ' ) . Since p€R, the points # and #' are uniquely determined, hence we can define a map g: (y,u) H* (x,u) from 2(K + pK') onto 2(/C) and a map ft: (y,u) fc^* (#', p_ 1«) from 2(7^ 4- p# ' ) onto 2 ( / 0 - It is easy to show that these maps are continuous, hence 17 * prf —g~l (7?) 0 h~l (17') is a Borel set. In [8] it was shown that v(SOn\R) = 0. This proves Lemma (3.1).

(3.2) Theorem. / / KfK' G K" are convex bodies and T? C X(K), T?' C S(K') are Borel sets, then c

(3.3) f Qi(K + pK\ri*pv,)dp(p)=io(1

k)@k(K)r1)Bi-k(K'fvt)

Json

for 1 = 0,...,» — 1.

The proof uses the connection of such formulae with contact probability problems, as treated, e.g., in [6], [8], [16]; these will not be considered here (compare also the survey article [14], in particular p. 159).

267

Let K,K'eiC be given. By r(K, K') = min {|| x - x' \\\ x G K, x' G K'} we denote the Euclidean distance of K and K\ and for r(K,K')>0 we write u(K,K,)i=(x,x')/r(K,K') if xeK, x'eK' are such that ||A?-*'ir= = r(K, K'). For r?, r?' G $ ( 2 ) and for e > 0 define

NeiK,^^,^):^ {g£Gn: 3xeK,x' <EgK' s.th.

0 < ||* - * ' | | = r(/C,£tf') < e, (x,u(K,gK')) G T?, (*', -%u.(K,gK')) egV'} .

By an obvious modification of the proof of Lemma 2 (first part) in [8] one shows that Ne(K, K'\ 17, T/) is measurable. By Fubini's theorem we have

(3.4) M(Ne(tf,^W))= J J d\ndp(p) Json JT(P)

with

np)={teE«:rtp eNe(K,K'-iV,r)')} .

Suppose that t&T(p). Then there exist points xGK, x' G pK' 4- t such that

0<\\x-x'\\ = r(KtpK' + t)<e,

(x} u(K, pK' + £)) G T? , (#', -u(K, pK' + t)) Gpr)' + t.

Write x' = py' 4- £. The vector u — u(K, pK' +1) clearly satisfies (x,u) G G 2 ( / 0 and « - « ) G S ( p ^ + f), thus

(* ,H)G7?n2(/0 and ( p / , - « ) Gp [r?' O 2 (/C')] .

Writing r = r(K, pK' +1), we have x' = x 4- ru and hence t — x - py' + + ru. It is obvious that p(K-pK', t) = x - py' and u(K- pK', t) — u, thus

{p(K-pK\ t), u{K~pK\ t)) = ix-py', u) G

G iv n s (#)] *(-p[r? 'ns (#')]),

which shows that

(3.5) £ GAfe(tf-ptf', [r? n 2(*0] * (-p [r?' fl Z(/C')])) .

If we now assume that pER (as defined in the proof of Lemma (3.1), but now for the bodies K and — K'), then we can reverse the argument and conclude that (3.5) implies t G 7\p). Thus we have

T(p)=MJK-pK', In r)X(K)] * (rpW nS(K')])) for p GR .

268

From (3.4) and (2.1), which can be applied because of Lemma (3.1), we deduce that

-•i-"zV"'(i) / &i(K~PK'> fonz(J0]«(-Pl»»'ni(jOi))<fr(p).

On the other hand, we have

lx(Ne(K,K';n,n')) =

= ^- 2 (?)e' "i\n7i)®k(.K,n)8n-i-k(K',v') •

With the obvious modifications, the proof is the same as that of equality (*) in [8], p. 9. The weak continuity of the measures 0,-, which is required in the proof, was shown in [9], (2.2).

Now we choose Borel sets rj C2(/0> rf'CX(K') and compare the coefficients of en~% in both representations of n(N€(K,K'; 17,17'). Replacing K' by —K1 and r?' by -r?', we arrive at formula (3.3). The proof of Theo­rem (3.2) is complete.

From the case i = n — 1 of (3.3) we now derive an integral-geometric formula which looks more complicated, but will be useful in an application to be treated elsewhere.

For an ^-dimensional convex body KG K? and for x € E n , we define v(K,x) as the unique exterior unit normal vector to K at x if x is a reg­ular boundary point of K\ otherwise, v(K,x) is not defined. IA denotes the indicator function of A.

(3.7) Theorem. Let K,K' G K" be n-dimensional convex bodies, let j3 CbK and P' CbK' be Borel sets, and let f,g : Sw_1 -* R be nonnegative, measur­able functions. Then

f h+pp(x)f{v(K+pK', x))g(p-lv(K+pK', x))dCn^{K + pK\ x)dv{p) Jso„ J

'n

n-1 = * ? 0 ( W * 1 ) ( / / f l W A , , ) i e * ( ^ ( * » « ) ) ) (\ty(x)8Wd®n-X-k(K\{x,u)))

269

Before the proof, a few remarks are in order. For a convex body K with interior points one has CH-l(Ktfi) = Hn"i(bKnf)J where Hn~l is the (n— l)-dimensional Hausdorff measure. Thus the inner integral on the left side extends over d(K + pK') and is with respect to the ordinary area meas­ure. Since the set of singular boundary points of a convex body is a Borel set of area measure zero, the function x *—> v(K + pK't x) is Cn-i(K + pK', •)-almost everywhere defined, and on its domain it is continuous. Hence, for ^-almost all p the integrand of the inner integral is (almost everywhere de­fined and) measurable. A remarkable aspect of the above formula is the fact that on the left side only the ordinary area measure occurs, while on the right the generalized curvature measures are indispensable to express the result.

Proof of Theorem (3.7). Suppose that KtK\p,p',f,g satisfy the assump­tions. Let OJ,G/ C 5 W _ 1 be Borel sets. First we remark that

JW< (3.8) JIfj(x)Ico(v(K,x))dC„-1(K,x) = &n-1(K,pxo>) .

In fact, by formula (4.3) in [11] we have

©„_ , (# , p x co) = H"" 1 0r , (2 ( *0 n (fi x to))) ,

where itl: E" x S"'1 -* E" is the projection onto the first factor. Now, x £irl(2(K) H (fi x co)) if and only if x€ /3 and (# ,« )€ 2(if) for some u 6 to. If x is regular, the latter is equivalent to v(K,x)Gw. Since H"_1-almost all boundary points of K are regular, we get

0„ . I (K- ,^xco)=H" _ 1 ({»e |3 :»( i f ) x)eco}) =

= \l&(x)I0i(v(K,x))dH"-l(x) ,

which proves (3.8). From (3.8) (applied to K + pK' etc.) we get for p G S0„

h+pp>MI0}(v(K + PK', *))Jw0>-»e(*+ PK\ *))iC„_1(K + pjr , *) =

= Jle+p^(x)Icjnpcjl(v(K + pK',x))dCn.1(K + pK',x) =

= Q^iK + pK', (0 + P/3') x (co npco')) =

= el,-l(K + pK', (flx co)*p(/3' x co')) .

270

Integration over S0n and application of (3.3) shows that the formula of Theorem (3.6) holds in the special case where / = / w and g = I(jJt. By linearity the result extends to finite linear combinations of indicator functions of spherical Borel sets and then by monotone convergence to arbitrary non-negative measurable functions / and g.

4. A short proof of formula (1.1)

For the proof of (3.6) we referred to [8], and there the crucial result was deduced from formula (1.1) above, which was proved in [6]. The proof given there used a slightly complicated characterization theorem obtained in [5]. In the following we give a direct proof of (1.1). It suffices to consider the case i = n — 1, since the general case is then easy to get (cf. [6], §4). Let us first show that

(4.1) f Sn-^P + pP', conpa>')dHp) = n^in~kl)Sk(P,u)Sn.^k{P\^)

Json

for convex polytopes P,P' G Kn and Borel sets to, OJ' € P ( S W _ 1 ) .

Forgiven pGS0„, let F bean (n — l)-face of the polytope P + pP* and u its exterior unit normal vector. Then F = G + pG\ where G is the face of P in which the supporting hyperplane to P with exterior normal vector u intersects P, and pG' is defined in the same way for pP'. If dim G = i and dim G' == k, we may assume that i + k = n — 1, thus ex­cluding only a set of rotations of ^-measure zero (for a simple proof, see Lemma (2.1) in [4]). Denoting by a(P,F) the set of exterior unit normal vectors to P at its face P, we have uGa(P,G) H o(pP',pG'). Vice versa, if G is an f-face of P and Gf is an (n- l - / ) - face of P' with o(P,G) H no(pP', pG,)¥=0, then G + pG' is a /-face of P + pP' with ; < n - l . Again excluding a set of rotations of measure zero, we may assume that ; = n — 1. Hence, for almost all p,

5 w . 1 (P4-pP ' , o ;npo ; ' )= S \n~1(F) = F G T „ - I ( P + P P ' )

o(p+pp•', F) n co n pco '*0

= "]£ 2 2 X^UJF + PP''), *' = 0 FGfyP) F'GF„-i-i(f')

v —, '

a(p,F) n a(pp', PF') n to n pco'* «

271

where F,(P) denotes the set of /-faces of P. Observing that, for the rota­tions and faces considered, the set

A(p,F,F') := [o(P,F) H <o] H p [o(P\ F') D OJ']

is either one-pointed or empty, we arrive at

f Sn-1(P + pP',unput)dHp) = Json

= "s 1 Z 2 / y-HF + pF') card A(p,F,F')dv(p).

Let i € { 0 » - l } , FETi(P), F' G ?„_!_,(?') be fixed. For almost all p we have dim(F + pF*) = n~ 1 and hence

Xn-1(F + pjP') = D(p)V(F)rX , ,-1"'(F'),

where the factor D(p) depends only on the relative position of the linear subspaces L(F),L(pF') parallel to F,pF\ respectively, and hence only on p. It remains, therefore, to compute

f D(p) card ( [ a ( P , F ) n w ] n p [ a ( P ' , ^ ) n c o ' ] ) ^ ( p ) . Json

(An argument similar to the subsequent one was used in Schneider-Weil [13], § 6). More generally, we consider the integral

7(j3,j3'): = f D(p) card (j8 n P j 8 ' ) ^ (p ) JsQn

for arbitrary Borel sets fl C s""1"' := L(F)L nsn~l, 0' C s1' := L ^ ' ) 1 n S*"1

(•*• denotes orthogonal complement). The measurability of the integrand is easy to see. If TESO„ is a rotation which maps sn~x~% into itself and keeps the orthogonal subspace pointwise fixed, then D{jp) — D{p) and hence f(r$,P') = /(j3,j3') by the invariance of v. Since f(',P') is obviously non-negative, finite, and a-additive, it follows from the uniqueness of spherical Lebesgue measure that /(j3,j3') = const •aw~1~'(j3), where the constant depends on /3'. Together with a similar argument, where the roles of |3 and f}' are interchanged, this shows that /(j3,/T) = a„jOn~1~i(fi)ot(fl') with a po­sitive constant an-t, which evidently depends only on n and i. Using this

272

with P = o(P,F) fi co, 0' = a(P', F') n co' and recalling that

1 ^e^(p)

we arrive at

Json

sn.1(P+pP\^nP^)dp(p) = ni1bnisi(Pt^)Sn.^i(P,>^)

* = 0

with positive constants b„i.

The extension of this formula to general convex bodies K, K' is easily done by means of an approximation argument analogous to the one used in [12] (Lemma (1.3); for assumption (b) of that lemma in the present case see [6], (3.5) and (3.6)). Finally, the explicit values of the constants are deter­mined by choosing balls of radii 1 and r for K,K' and co = co'= Sw_1 , and then comparing the coefficients of equal powers of r.

5. A projection formula

From formula (3.3) we shall now deduce a representation of the gener­alized curvature measure as a rotational mean of curvature measures of pro­jections.

If E C En is a linear subspace, we denote by x | E the image of x G En

under orthogonal projection onto E, and we use a similar notation for sub­sets of En. For i | C 2 we define

rj | E : = {(x | E, u) : {x, u) G17 and u G E} .

(5.1) Theorem. Let K€Kn be a convex body, rj C 2(K) a Borel set and ECEn a q-dimensional linear subspace. Then

(5.2) f ®j(K\ pE, n I pE)dv(fi) = qKq ®j(K,n) Json

for ^ E {1 , . . . , if - 1}' and j G {0,. . . ,q — 1}, where &• is the generalized curvature measure taken with respect to the subspace E.

For the special case of the surface area measures Sy, such a result was

273

first proved in [6]. This has found applications in Schneider [6], Goodey-Schneider [4], Goodey [3]. For the curvature measures C;, a corresponding formula was obtained by Weil [16]. In neither case it was observed that the projection formula is an almost immediate consequence of the Minkowski sum formula.

To see this, we choose an (n -q)-dimensional convex body K' with •tfi-q(#') = i and OEK' and we choose r?' as the set of all support ele­ments (x,u) of K' with x in the relative interior of K'. It is then clear from (2.1) that

. nKa( )" for ; = n — ch (5.3) 0,(if, „') = \ «V

for j =£ n - q .

For given e > 0 , let xEM€(K + K\ r\ *r?'). This means that 0 < <r(K + K\ x)<e and (p(K + K't x),u(K + K',x))er}*'n'; in particular, u :=u(K + K',x)eri', hence « G £ , and p(K + K', x)= y +y' with (y,u) E17 and (y', u') E77'. This implies (y | E, u) E r\ | E. Obviously we have \\x\E-y\E\\ = r(K + K',x), p(K\E,x\E)=y\E, and u(K\E, x I E) = w, thus # IJE" EAfg(/ir| i?, 771 E), where M'e is taken with respect to the subspace E. Vice versa, one easily shows that each point in Alg(X| E, rj\E) is the projection of a point in Me(K + K', 17 * T?'), thus

MJK + K', r\ * 7?') I £ = M'JKl E, r\ I E).

If K and /£' are not in special relative position, then the set of points in Me(K + Kr, 77 * 77') which project into the same point of E, is a translate of the relative interior of K'. Hence, Fubini's theorem gives

Xn (Me(K + K',n* 77')) = X« (M'e(K IE, r? | E)),

and (2.1) yields

n i=o l, q j = o ]

For ^-almost all p, this can be applied to pK,pr]',pE instead of K\ 17', E. If we then integrate over all rotations p with respect to the Haar measure v, use (3.3) and (5.3) and compare the coefficients of equal powers of e, we get formula (5.2).

274

BIBLIOGRAPHY

[I] W.J. Firey, Kinematic measures for sets of support figures, Mathematika 21, 270-281 (1974).

[2] W.J. Firey, Inner contact measures, Mathematika 26, 106-112 (1979).

[3] P.R. Goodey, Limits of intermediate surface area measures of convex bodies, Proc. London Math. Soc. (3) 43, 151-168 (1981).

[4] P.R. Goodey, R. Schneider, On the intermediate area functions of convex bodies, Math. Z. 173, 185-194(1980).

[5] R.Schneider, Kinematische Beriihrmafie fur konvexe Korper, Abh. Math. Sem. Univ. Hamburg 44, 12-23 (1975).

[6] R. Schneider, Kinematische Beruhrmafie fur konvexe Korper und Integralrelationen fur Oberflachenmafie, Math. Ann. 218, 253-267 (1975).

[7] R. Schneider, Curvature measures of convex bodies, Ann. Mat. Pura Appl. 116, 101-134(1978).

[8] R. Schneider, Kinematic measures for sets of colliding convex bodies, Mathematika 25,1-12(1978).

[9] R. Schneider, Bestimmung konvexer Korper durch Krummungsmafie, Comment. Math. Helvet. 54, 42-60 (1979).

[10] R.Schneider, Boundary structure and curvature of convex bodies, In: Contribu­tions to Geometry, eds. J. Tolke and JJVI. Wills, Birkhauser Verlag, Basel etc. 1979, pp. 13-59.

[II] R.Schneider, Parallelmengen mit Vielfachheit und Steinerformeln, Geom. Dedicata 9,111-127(1980).

[12] R. Schneider, Curvature measures and integral geometry of convex bodies, Rend. Sem. Mat. Univ. Politecn. Torino 38, 79-98 (1980).

[13] R.Schneider, W.Weil, Translative and kinematic integral formulae for curvature measures, Math. Nacbr. 129, 67-80 (1986).

[14] R.Schneider, J.A. Wieacker, Random touching of convex bodies, In: Stochastic Geometry, Geometric Statistics, Stereology, eds. R. Ambartzumian and W. Weil, Teubner Verlag, Leipzig 1984, pp. 154-169.

[15] W.Weil, Beruhrwahrscheinlichkeiten fur konvexe Korper, Z. Wahrscheinlichkeitsth. verw. Geb. 48, 327-338 (1979).

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[16] W. Weil, Kinematic integral formulas for convex bodies, In: Contributions to Geometry, eds. J. Tolke and J.M. Wills, Birkhauser Verlag, Basel etc. 1979, pp. 60-76.

ROLF SCHNEIDER - Mathematisches Institut, Universitat Freiburg i. Br.

Lavoro pervenuto in redazione il 3/VI/1985