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W . R O T H E R A N D M. Z , ~ H L E
A B S O L U T E C U R V A T U R E M E A S U R E S , I I
ABSTRACT. The absolute curvature measures for sets of positive reach in R d introduced in [7] satisfy the following kinematic relations: Their integrated values on the intersections with (or on the tangential projections onto) uniformly moved p-planes are constant multiples of the corresponding absolute curvature measures of the primary set. In the special case of convex bodies the first result is the so-called Crofton formula. An analogue for signed curvature measures is well known in the differential geometry of smooth manifolds, but the motion of absolute curvatures used there does not lead to this property. For the special case of smooth compact hypermanifolds our absolute curvature measures agree with those introduced by Santal6 [4] with other methods.
In the appendix, the section formula is applied to motion invariant random sets.
1. E Q U I V A L E N T D E F I N I T I O N S
The present paper is a continuation of [7], which also contains motivations
and references to the related literature. We first recall some notions introduced there. At the same time we correct an error in formulas (3)-(6) of [7], where certain projection Jacobians have to be inserted.
In the sequel, X denotes an arbitrary subset of R d of positive reach (in the sense of Federer [1]) and B a bounded Borel set in R d x S d- 1 (or in Rd). Note
that convex bodies and compact C2-submanifolds are special cases for X. In [7] the kth absolute curvature measure c~bs(x, B), k = 0 . . . . . d - 1 , is intro-
duced by means of the natural invariant measure of the set of all ( d - 1 - k)- planes locally colliding with X inside B. Thereby affine and linear subspaces of R d are considered as dements of sufficiently large Euclidean spaces via
representation by multivectors. Let nor X be the unit normal bundle of X and G(X, d, k) its kth Grassmann bundle (denoted in [7] by G(X, k)), i.e.
G(X, d, k )= {(x, n, V): (x, n)~nor X, V ~ G , ( d - 1 , k)}
where G,(d-1 , k) is the Grassmann submanifold of G(d, k) of those k- dimensional linear subspaces of R d which are orthogonal to the unit vector n ES d-1. Let v" be the normalized rotation invariant measure on d - l , k
G , ( d - l , k ) . 17 v is the orthogonal projection onto the subspace V The function
f (x , n, V) = (IIv~ x, V)
maps G(X, d, k) onto the set sO(X, d, k) of parametrized k-dimensional affine subspaces (k-planes) locally colliding with X. Let P be the coordinate
Geometriae Dedicata 41: 229-240, 1992. © 1992 K l u w e r Academic Publishers. Printed in the Netherlands.
230 w . R O T H E R A N D M. Z A H L E
projections in d ( X , d, k) given by
P(y, v ) = v.
~ s is the s-dimensional Hausdorff measure in Euclidean space and Js the s- dimensional Jacobian function of a locally Lipschitz mapping in the sense of Federer [2].
For m = d - 1 + k ( d - 1 - k) the weighted Hausdorff measure
(1) gk(X, "):----= gd.k 1 I l(.)(y, V)Jk(a_k)P(fl , V)~'f'(d(y, V)) d ~(x,d,k)
is the natural motion invariant measure of k-planes locally colliding with X. (Cf. (1) in [7], corrected. The constant gd,k is defined there.) Using the co-area formula for the mapping P one obtains
.)) = ~ ~ d - 1 - k(d(X ' d, k) c~ P - I(V) ~ ('))vd,k(d V) (2) g k ( X , j~ (d,k)
where Va,k is the normalized motion invariant measure on G(d, k) (cf. (2) in [7] corrected). On the other hand, the co-area formula applied to the function f yields
#k( X, ") = 9a,~ f l(.)(f(x, n, V))Jmf(X , n, V)Jk<d_k) 3~ (x,a,k)
x P(f(x, n, V))~"(d(x, n, V))
since by Sard's theorem f is a.e. bi-unique. Therefore, the measure
(3) c~b_sl _k(X,B):= Cd, k f 1B(X, n)J.,f(x, n, V)Jktd_k)P(f(x , n, V)) d~ (X,d,k)
x ~¢gm(d(x, n, V))
= const ~k(X, f(G(X, d, k) c~ B x G(d, k)))
may be interpreted as the invariant measure of k-planes locally colliding with X inside the location-direction set B c R a x S a- 1. (Cf. (4) in [7] corrected. The constant Cd.k is defined there by certain normalization.) c~bs(x, ") is also said to be the kth absolute curvature measure of X in view of the following considerations.
Applying the co-area formula to the coordinate projection ~: G(X, d, k )~ nor X we get
CabStYB)=(a-~l)~r(k)-l~na - 1 - kl, xx ' or Xc3B f G n(d - 1,k) Jd-17z(x'n'V)-~
x Jmf(X, n, V)Jk( d_k)P(f(x, n, V))v~_ 1,k(dV)J~ a- l(d(x, n)),
A B S O L U T E C U R V A T U R E MEASURES, II 231
where a(k)=~fk(S k) (cf. (5) in [-7], corrected). The expression under the integral may be calculated as follows. Any V~ G,(d, k) may be represented by
V = (V' ^ n)"
where V' is the (k-1)-space orthogonal to V and n. (W A W means the subspace of R e generated by the subspace W and the vector w, i.e. the corresponding unit multivector with identification of the signs.) In this notation the differential of f acts on the tangent space to G(X, d, k) at (x, n, V) through
Df(x , n, V)(O, O, u) = (0, u),
u ~ TvG,(d- 1, k),
and
Df(x , n, V)(v, w, (V' A Hvw) ±)
= (IIviV+(X, n ) I l v w + ( x , Ilvw)n, (V' A IlvW)-t),
(v, w)e Ttx,, ) nor X. (Here (a, b) means the scalar product of the vectors a, b.) Since the vector (x, n )Hvw+ (x, rlvw)n is orthogonal to I Ivw and equals 0
if Df(x, u, V)(v, w, (V' /x Hvw)±)skerDP(f(x , n, V)) (i.e. if Hvw = 0) we conclude that
Jk(a-k)P(f(x, n, V))J,,f(x, n, V) = [I A k(a_k)Pvl[ 11 A , , fv II
where the linear mapping fv is defined on T(x,,,v)G(X, d, k) by
iv (0, O, u) = (0, u), u E Tv G, (d - 1, k)
fv(v, w, (V' /x rIvw) ±) = (Hviv, (V' A rlvw)±), (v, w)s T(x,,)nor X,
and Pv is defined on the image space fvT~ .... v)G(X, d, k) by
Pv(z, W) = W.
(fv and Pv correspond to Df(x , n, V) and DP(f(x, n, V)), respectively, when the vector (x, n) I Ivw+ (x, Hvw)n from above is neglected.) Next one easily verifies that the right-hand side of the last relation equals
Je_~z(x, n, V)Je-~hv(x, n),
where hv is given on nor X by
hv(x, n) = (rlv~X, rIvn),
i.e.
Dhv(x, n)(v, w) = (Hv~v, Hvw).
2 3 2 w . R O T H E R A N D M. Z . ~ H L E
Hence,
Jd_l~(X, n, V)-lJmf(x, n, V)Jk(d_g)P(f(x, n, V))= Jd-lhv( x, n)
(cf. (6) in [7], corrected). The computation OfJd- xhv(x, u) (cf. [7]) leads to the following curvature representation of c~,bs(x, B): In an earlier paper it is shown that for almost all (x, n) s nor X there exist d - 1 generalized principal curvatures xl(x, n) . . . . , Xd- I(X, n) e ( - - 0% + oo] and directions of principal curvatures bl(x, n), . . . , be-l(x, n) which form, together with the unit normal n, a positively oriented orthonormal basis in R d. Then one infers the final formula
(4) C~,_k(X,B)=(d;1)a(k)-lfnorX~BfC..(d_~,k) d - 1
× 1--[ (1 + tee(x, n))- 1/2 E Xi(x)(X, n)... Xi(k)(X, n) j=l i(1)< -.- <i(k)
x IHvbi(1)(x, n) ^ ... ^ Hvbi(k)(X, n)l 2 v~_ a,k(dV)~ d- ~(d(x, n)).
If some of the ~cj are equal to 4- ~ then the product and the sum under the integrals are formal and the corresponding limits as xj ~ ~ are meant. (Cf.
(8) in [7].)
REMARK. Omitting the absolute value sign under the integrals we obtain the classical signed Lipschitz-Kill ing-Federer curvature measures in the case when 1B(x, n) does not depend on the direction n. For simplicity we consider in the sequel only such B identifying them with subsets of R d and using the
same notations.
2. RESTRICTION TO SUBSPACES
Any p-plane A in R d will be parametrized as before by the pair (z, W) such that A -- z + W where W ~ G(d, p) (p-dimensional Grassmanian of R d) is the
direction space of A and z = H w i x for arbitrary xEA. Let Vd,p be the normalized rotation invariant measure on G(d, p). Then we introduce the motion invariant measure Ctd, v on the parameter space of p-planes d (d , p) by
O:d,p=fG(d,p) fwllt '(z 'W)dzv'~,v(dW)"
(For Lebesgue measure dz on W i we also write ~d-P(dz) and sometimes we identify the pair (z, W) with the plane A = z + W..) For ~e,p-almost all
A B S O L U T E C U R V A T U R E M E A S U R E S , II 233
A e d ( d , p ) the section set X n A is also of positive reach (cf. Federer [1]). Therefore, its absolute curvature measures C~b~(X n A, B) are determined. Our notions are correct in the following sense:
T H E O R E M 1. The absolute curvature measure c)bs(x n A, B) of the set X n A c R d agrees with the j th absolute curvature measure of X n A c A
(restricted to B n A) calculated with respect to the p-dimensional reference space A if j <<. p - 1.
Proof Recall that
-1,(X, B) = Ca,t, ,I~¢[ (X,B,d,k)Jk(a-k)P(Y, V)jt~m(d(Y, V)) C~1~1
where
and
~¢(X, B, d, k) = f (G(X , d, k ) n B x S d- I x G(d, k))
P: d ( X , d, k) --. G(d, k)
is the coordinate projection P(y, V)= E W e n o w ident i fy the p -p lane A = z + W w i t h the E u c l i d e a n p-space R p a n d
consider X n A as a subset of R p. Then the variant of f is the mapping
f,~ : G(X n A, p, k) ~ d ( X n A, p, k)
given by
f~(x, r~, P) = ( n ~ H w x , ~).
Analogously, P~v: d ( X n A, p, k) --, G(p, k) denotes the corresponding coor- dinate projection. Put j = p - 1 - k . In this notation it suffices to prove that
c(d, k + d - p) [ J(k + d- p)(p - k)P(Y, V)J'~q(d(Y, V)) J ~¢(XnA,B,d,k + d-p)
k) [ Jk(p-k)P~r@, [z)~'a'(d@, V)) c(p, (XnA,BnA,p,k)
where
d ( X n A , B n A , p, k ) = f ~ c ( G ( X n A , p, k ) n B n A x S p-1 x G(p, k)),
q = d - 1 + ( k + d - p ) ( p - 1 - k ) ,
and
r = p - l + k ( p - l - k ) .
234 w . R O T H E R A N D M. ZA.HLE
For, we introduce the auxiliary set
S={(~, V,, I~): (~, ~ / ) • ~ ( X n A , B n A , p, k),
IYVeG(d, d - p ) , W± V,, ~ ' n W = {0}}
(where A = z + W as before) and the mapping q~:S--* ~ ¢ ( X n A , B, d,
k + d - p ) by
~0(y, ~ v?)= (n(~ea,)~ y, ~ ® ~'). In this case the co-area formula applied to (p yields
Jq(p(y, V, W)J(k+d_p)(d_k)P((p(y , V, W))Jt~g(d(y, V, W))
= ~ J(k + d - p)(p - k) P(Y, V)~Vt~q(d(Y, V)). d d (XnA,B,d,k + d - p)
(For fff~-almost all (y, V)• d ( X n A, B, d, k + d - p ) the pullback (p- l(y, V) is a singleton.) Further, by construction we have
J(k+~-~(~-k)P((p(Y, v, fie))= Jk(~-,)P~(Y, v), a.e.
Let n here be the coordinate projection on S given by
~(y, ~, ffz)= (y, ~).
Then the integral on the left,hand side of the above equation agrees with
fd J,(~ k)P~(Y,~') f( Jq(P0~, V,, ~/) (XnA,BnA,p,k) I~: (~, V,,I~z) e S}
x J~(~, V,, lTV)~/t~q-~(dl~)Jf'(d(~, V)).
The inner integral is a constant which is equal to c(p, k ) c ( d , k + d - p ) - ~
(This value may be obtained, for example, by considering the well-known convex case.) Comparing the entire integral expressions one obtains the assertion. []
3. I N T E R S E C T I O N W I T H M O V E D P L A N E S
As a consequence of the proof of Theorem 1 and (3) we get, for almost all W • G(d, p) and z • W ±, the relation
(5) r~abs _k(X n (z + W), B) = c(p, k) ~ la(x) ,,Fp_ 1 3~ (Xn(z + W),p,k)
x J,f~v(X, fi, - z z v)Jk(~_,~Pw(fC~(x, ~, ~)w'(a(x, ~, ~)). This enables us to prove the following kinemat ic sect ion formula .
ABSOLUTE C U R V A T U R E MEASURES, II 235
THEOREM 2.
f d abs A B - Cp_ 1 - k( X n , )•d,p(dA) - y(p, d - 1 - k, d)C3~ 1 _k(X, B). (d,p)
Proof. It suffices to show that the integration of (5) with respect to W and z yields (3) up to a constant. For, we introduce the auxiliary space
M = x, IHwnl ' rIw~('IIv' w):(x, n, V)eG(X, d, k), WeG(d, p),
I Iwn ~ O, dimHw~{,}~ V = k}
and the mapping
y(x, ~, P, w) = (n~x , ~, w)
defined on M. j7 may be split into two orthogonal parts: since ~" c W, we have
(6) (lI~x, P,, w ) = (rI~4nwix+rIwx) , P, w)
= (rlw~x, O, W) + ( I I ~ I I w x , P,, O)
and the last two vectors are orthogonal. (Recall that ~" and W are considered as elements of certain Euclidean vector spaces.) Let g: M ~ d(d , p) be given by
g(x, ~, ~ w) = (nw~x, w).
Then g-l(z, W) may be identified with the set G ( X ~ (z + W), p, k) where the last component W is omitted. Further, for Hw~x=z we have
(II9A-Iwx, V, 0) = (f~v(X, fi, V), 0).
In this case (6) yields the following relationship between the Jacobians:
J~f(x , fi, V,, W) = J(o+a)(d-p)g(x, fi, V,, W)J~f~v(X, fi, V)
where ~ = ( p + 1)(d-p)+r and, as before, r = ( p - 1 ) + k ( p - 1 - k ) . From this and the co-area formula applied to the mapping g and then to
the coordinate projection ~: d (d , p) ~ G(dl p) with re(z, W) -- W we infer
flu lB(x)jeaf(x ' ~,, . . . . . . nw~x, ,nw~x, fi, vv )ak(p k)r w tJw ~x, fi, V))
× J.(._.)~(a(x, ~, P, W))~e~(d(x, ~, P, W))
=conSt fa(d,p) fW~ fa(ic~(z+W),p,k) l '(x)J'f(v(X' fi' P)
X 4(p- k)P~v (f(v (x, fi, ~'))2/g d- P(dz)vd,p(d W).
236 W. R O T H E R A N D M. Z ,~HLE
In view of (5) the last expression corresponds to the right-hand side of the assertion. We now will show that the left-hand side of the above equation agrees with C]~ 1 -k(X, B) up to a constant. For, define the auxiliary rotation mapping
h: G(X, d, k) x G(d, p) ~ M, a.e. by
Hw n h(x, n, V,, W ) = x, I-ff~wn[' nw~{.}~v, W)
whose ~-Jacobian equals 1. Taking into regard the relation
J , . f (x , n, V ) J k ( d _ k ) P ( f ( x , n, V)) IIw±x HW±X
=daf (h (x , n, V, W))Jktp-k)Pw ( fw (h(x, n, V, W))
x Jp(d_p~(goh(x, n, V, W))
which holds for almost all (x, n, V, W), we obtain by (3)
_ k(X, B) = const f l ts(X)Jmf(X, n, V)Jkld-k)P(f(x, n, V))~Cm(d(x, n, c ]bs 1 v)) J~ (X,d,k)
= const [ 1B(X)Jmf(X, n, V)Jk( ~_ k)P(f(x, n, 11)) 3~ (X,d,k) x Cr(d,p)
x ~ " + p~d- P)(d(x, n, V, W))
=const .fM 1B(X)J~f(X, fi, V,, . . . . . . nwix. pnwix, W)ak~p-k)rw tJw tx, fi, V))
x Jt,(a_t,)zt(g(x, fi, V,, W))jfk(a-l ')(h-l(x, fi, V,, W))jf~n(d(x, fi, V,, W))
according to the co-area formula for the mapping h. By construction, for almost all (x, fi, V,, W), aUfk(a-P)(h - l(x, fi, V,, W)) is a constant. Hence, the last expression agrees with the left-hand side of the former equation up to a constant, which may be determined by choosing a ball for X. []
REMARK. If X is a smooth compact hypermanifold in R d our absolute curvature measures agree with those introduced in Santal6 [4] up to certain constants. In this case Theorem 2 corresponds to the section formula (5.1) in [4] which is proved with classical differential geometric methods. Note that in distinction to Santal6's approach for lower-dimensional submanifolds our absolute curvature measures are determined for all orders and that they are the natural counterpart to-the signed-Lipsc_hitz-Killing-Federer curvature measures.
ABSOLUTE C U R V A T U R E MEASURES, II 237
4. P R O J E C T I O N ONTO MOVED PLANES
We now turn to the projection formula. In Schneider [5] the related result for convex bodies was extended to the convex ring by means of tangential projections. Here we will prove a similar relationship for sets of positive reach.
First recall the equivalent expression for the absolute curvature measures in terms of mean tangential projections: For any VE G(d ,d -1 -k )
H(X, B, V ±) := {y E Re: (y, V) E d ( X , B, d, d - 1 - k)}
is the set of orthogonal projections onto V ± of those x E X c~ B where x + V is a colliding plane. We write
H(X, V ±) := H(X, R d, V±).
In view of (2) we obtain
(7) ,bs _ f ~k( I I (X, B, V±))va,a_ 1 - k(d V) Ck (X,B)--ad,d-l-k ,)6 ( d , d - l - k )
for certain constant ad, d 1-k. (This formula establishes the relationship to Santalb's absolute curvature measures in the special case of smooth hypermanifolds.)
For an arbitrary WEG(d,p) we consider the tangential projections H(X, B, W) as subsets of the Euclidean p-space W and introduce the absolute
curvature measures C~bs(1-I(X, W), H(X, B, W)), 0 ~< k ~< p - 1, as follows: Denote G(W, k):= {VEG(d,k): V c W}. The set
d ( H ( X , W), H(X, B, W), p, p- - 1 - k) := {(y, U) E W e G(W, p - 1 - k):
(y, U • W ±) E d ( X , B, d, d - 1 - k)}
may be interpreted as the set of ( p - 1 - k) planes in W locally colliding with H(X, W) inside the Borel set H(X, B, W). Let P be the coordinate projection given on this set by P(y, U) = U. Then we define for r = p - 1 + k ( p - 1 - k)
C~,bs(H(X, W), H(X, B, W)=Cp, k f~(rI(X,VO,n(X,B,W),p,p l -k )
x Jv~k+ 1)P(Y, U)~r(d(Y, U))
which is compatible with the preceding notions and notations and with the variant in the convex case.
Let VW,k be the normalized rotation invariant measure on G(W, k). Similarly
238 w . R O T H E R AND M. Z A H L E
as in (6), one immediately verifies that
clb~(n(x, w), n ( x , B, W))
=ap,p_ l_k ~ ~gk({y:(y, U)ed(H(X, W), 36 (W,p- 1 - k)
r I (x , B, W), p, p - 1 - k})vw,~_ , _ k(dU)
which is equal to
-k f ~fk({yeRd: (y, U • W l) ap,p-1 JG (W,p l - k )
e ~¢(X, B, d, d - 1 - k)})Vw,p_l -k(dU).
Integrating the right-hand side over W ~ G(d, p) with respect to the measure vd,p one obtains in view of (7) the curvature measure C~b=(X, B) up to the
constant
Thus we proved the following projection formula:
T H E O R E M 3.
~G abs Ck (H(X, W), H(X, B, W))Vd,.(dW) (d,p)
= 7(p-- l , d - l - k , d-1)-Ic~,b=(X,B), 0 ~ k ~ p - 1.
4. APPENDIX
As an application we consider the absolute curvature measures of random sets with positive reach. (Corresponding results for Lipschitz-Kil l ing-
Federer curvature measures may be found in [6].) Let ~ be a random closed set of R d (in the sense of Matheron [3]) whose realizations a.s. have positive reach. Then similarly as in [6], it can be shown that C~,b=(4, ") is a random measure. If we assume additionally that 4 is motion invariant then the absolute curvature intensity measure ab= ECk (4, ") has the same property and by the uniqueness of Haar measure we obtain l:c~,b=(4, ' )=2~L d for Lebesgue measure L e and certain constant 2~E [O, oo] which is said to be the kth
absolute curvature intensity of 4. Let A be a p-dimensional affine subspace of R e and O ~< k ~< p - 1. Then
c~bs(~c~A, ") is a random motion invariant measure on A and
ABSOLUTE CURVATURE MEASURES, II 239
EC~,bs(~ n A, -) = ~kc~Aj~fP(( " ) ~ A) where the sectional absolute curvature
intensity 2k ¢~a does not depend on the choice of A. Theorem 2 yields the
following stereological relationship between these intensities.
T H E O R E M 4. Under the above conditions we have
2 ~ n a k + p - d = 7 ( p , k , d ) 2 k ¢, k > i d - - p .
Proof Let V be a p-dimensional linear subspace, B an arbitrary Borel set and x the H a a r measure on the rota t ion group SO(d) of R d with x(SO(d)) = 1. Regarding the expression
f . t ~ C~,b+sp-d(¢ n (g V + z), B n (g V + Z))ovfd-P(dz)x(dg) E O(cO JoY
we obtain by Theorem 2 and the mot ion invariance of 4, that it is equal to
7(P, k, d)Y_C~,bs(¢, B) = y(p, k, d)A~Ld(B).
On the other hand, we get by Fubini 's theorem
fso(d) fgv ~ ECk + p-d(¢ n (gV + z), n (gV + z))~ff a- P(dz)~:(dg) B
(" (" ~¢~(oV+z) ~eptu z))Wa-.(dz)K(dg) = l I " ~ k + p - d ~,t. Ut~, t~ ( o F "q- d SO(d) dgv-
and since ~ ( g v + ~ ) _ ~ a for all (g, V, z) we may continue with " ~ k + p - d - - ~ k + p - d
k + p - d O(d) V ± V + z
~ n A d = 2 k + p_ ~L (B).
Note added in proof We would like to thank a referee for ment ioning an
error in Section 4 of Par t I of this paper [7]. There the finiteness of r of the generalized principal curvatures has to be assumed.
REFERENCES
1. Federer, H., 'Curvature measures', Trans. Amer. Math. Soc. 93 (1959), 418-491. 2. Federer, H., Geometric Measure Theory, Springer, New York, Berlin, Heidelberg, 1969. 3. Matheron, G., Random Sets and Integral Geometry, Wiley, New York, 1975. 4. Santal6, L. A., 'Total curvatures of compact manifolds immersed in Euclidean space', Istit.
Nazion. Alta Matematika, Syrup. Matem. XIV (1974), 363-390. 5. Schneider, R., 'Curvature measures and integral geometry of convex bodies, III', Rend. Sem.
Mat. Univ. Politec. Torino 46 (1988), 111-123.
240 w . ROTHER AND M. ZAHLE
6. Z/ihle, M., 'Curvature measures and random sets II', Prob. Theory Rel. Fields 71 (1986), 37-58. 7. Z~ihle, M., 'Absolute curvature measures', Math. Nachr. 140 (1989), 83-90.
Authors' address:
W. R o t h e r a n d M. Z/ihle,
M a t h e m a t i s c h e Fakul t / i t ,
F r iedr ich-Schi l l e r -Univers i t / i t ,
0-6900 Jena,
G e r m a n y .
(Received, August 28, 1990)
