A Remark on Mixed Curvature Measures for Sets with Positive Reach

The existence of mixed curvature measures of two sets in R with positive reach introduced in [6] is discussed. An example shows that the non-osculating condition from [6] does not ensure the locally bounded variation of the mixed curvature measures. Further, some sufficient conditions for the local boundedness of mixed curvature measures involving absolute curvature measures are presented. For any two subsets X, Y ⊆ R with positive reach and r, s ∈ {0, 1, . . . , d− 1}, r+ s ≥ d, the mixed curvature measures Cr,s(X, Y ; ·) have been defined in [6] (where a different notation, Ψr,s(X, Y ; ·), has been used) as integrals of certain (2d − 1)-forms ψr,s over the joint unit normal bundle nor (X, Y ) = f(((norX × norY ) ∩R)× [0, 1]), where norX, norY are the unit normal bundles of X,Y , respectively, R = {(x,m, y, n) ∈ R : m+ n 6= 0} and f : (x,m, y, n, t) 7→ ( x, y, sin(1− t)θ sin θ m+ sin tθ sin θ n ) , θ ≡ ∠(m,n) ∈ [0, π]. Then, a translative integral formula involving these mixed curvature measures was proved ([6, Theorem 1]) under the ‘non-osculating assumption’ L({z ∈ R : ∃(x,m) ∈ norX, (x+ z,−m) ∈ norY }) = 0. (1) Supported by the Grant Agency of Czech Republic, Project No. 201/99/0269, and by MSM 113200007 0138-4821/93 $ 2.50 c © 2002 Heldermann Verlag 172 J. Rataj, M. Zähle: A Remark on Mixed Curvature Measures . . . It may, however, occur (as remarked by Joseph Fu in personal communication) that nor (X, Y ) has not finite H measure (since f is only locally Lipschitz) and the signed measures Cr,s(X, Y ; ·) may not be correctly defined (see Example 1 below). Therefore, it has been assumed additionally in [7] that |C̄|r,s(X, Y ; ·) is locally finite, 0 ≤ r, s ≤ d− 1, r + s ≥ d, (2) where the (nonnegative) measures |C̄|r,s(X, Y ; ·) are defined below. Under this assumption, Cr,s(X, Y ; ·) is well defined for any admissible r, s and |C̄|r,s(X, Y ; ·) is the total variation measure of its projection C̄r,s(X, Y ; ·) = Cr,s(X, Y ; · × S) (cf. [6, Theorem 2], [7, Theorem 4.2]). The functionals C̄r,s(X,Y ; ·) were studied more extensively for convex bodies X, Y (or sets from the convex ring), see e.g. [9, 10]. Note that in [3], the notion ‘mixed curvature measure’ has been used for a different functional. Let κi ≡ κi(x,m) (λi ≡ λi(y, n)) be the (generalized) principal curvatures and ai ≡ ai(x,m) (bi ≡ bi(y, n)) the corresponding (generalized) principal directions of X (Y ) defined at H-almost all (x,m) ∈ norX ((y, n) ∈ norY , respectively). We set for any bounded Borel subset A ⊆ R |C̄|r,s(X,Y ;A) = ∫ (norX×norY )∩R 1A(x, y)Fr,s(θ)|W X,Y r,s (x,m, y, n)|H (d(x,m, y, n)), (3) where Fr,s(θ) = O −1 2d−1−r−s θ sin θ ∫ 1 0 ( sin(1− t)θ sin θ )d−1−r ( sin tθ sin θ )d−1−s dt,