Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 43 (2002), No. 1, 171-179.

A Remark on Mixed Curvature Measures for Sets with Positive Reach

J. Rataj^{1} M. Zähle

Charles University, Prague, Czech Republic Friedrich-Schiller-University, Jena, Germany

Abstract. The existence of mixed curvature measures of two sets in ℝ^{d} 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^{d} with positive reach and r, s ∈ {0, 1, …, d − 1}, r + s ≥ d, the mixed curvature measures C_{r,s}(X, Y; ·) have been defined in [6] (where a different notation, Ψ_{r,s}(X, Y; ·), has been used) as integrals of certain (2d − 1)-forms ψ_{r,s} over the joint unit normal bundle

nor (X, Y) = f(((nor X × nor Y) ∩ R) × [0, 1]),

where nor X, nor Y are the unit normal bundles of X, Y, respectively,

R = {(x, m, y, n) ∈ ℝ^{4d} : m + n ≠ 0}

and

f : (x, m, y, n, t) ↦ (x, y, \frac{sin(1 − t)θ}{sin θ}m + \frac{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^{d}({z ∈ ℝ^{d} : ∃(x, m) ∈ nor X, (x + z, −m) ∈ nor Y }) = 0.  (1)

1Supported by the Grant Agency of Czech Republic, Project No. 201/99/0269, and by MSM 113200007 0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 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^{2d−1} measure (since f is only locally Lipschitz) and the signed measures C_{r,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, C_{r,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 C̄_{r,s}(X, Y; ·) = C_{r,s}(X, Y; · × 𝕊^{d−1}) (cf. [6, Theorem 2], [7, Theorem 4.2]).

The functionals C̄_{r,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 κ_{i} ≡ κ_{i}(x, m) (λ_{i} ≡ λ_{i}(y, n)) be the (generalized) principal curvatures and a_{i} ≡ a_{i}(x, m) (b_{i} ≡ b_{i}(y, n)) the corresponding (generalized) principal directions of X (Y) defined at H^{d−1}-almost all (x, m) ∈ nor X ((y, n) ∈ nor Y, respectively). We set for any bounded Borel subset A ⊆ ℝ^{2d}

|C̄|_{r,s}(X, Y; A) =
∫_{(nor X×nor Y)∩R} 1_{A}(x, y)F_{r,s}(θ)|W_{r,s}^{X,Y} (x, m, y, n)|H^{2d−2}(d(x, m, y, n)),  (3)

where

F_{r,s}(θ) = O_{2d−1−r−s}^{−1} \frac{θ}{sin θ} ∫_{0}^{1} (\frac{sin(1 − t)θ}{sin θ} )^{d−1−r} (\frac{sin tθ}{sin θ} )^{d−1−s} dt,

W_{r,s}^{X,Y} = ∑_{|I|=r} ∑_{|J|=s} \frac{∏_{i∉I} κ_{i} ∏_{j∉J} λ_{j}[ ⋀_{i∈I} a_{i}, ⋀_{j∈J} b_{j}]^{2}}{∏_{i=1}^{d−1} \sqrt{1 + κ_{i}^{2}} ∏_{j=1}^{d−1} \sqrt{1 + λ_{j}^{2}}}

(the summation is taken over all subsets I, J ⊆ {1, …, d − 1} of given cardinality), O_{k} is the k-dimensional measure of the unit sphere in ℝ^{k+1} and [ ⋀_{i∈I} a_{i}, ⋀_{j∈J} b_{j}] is the Jacobian of the orthogonal projection of the linear subspace spanned by {a_{i} : i ∈ I} onto the orthogonal complement of that spanned by {b_{j} : j ∈ J}. Note that the measure C̄_{r,s}(X, Y; A) can be represented as in (3) with the difference that the absolute value in the integrand is missing (see [7, Theorem 4.2]).

Example 1. There exists a compact set X ⊆ ℝ^{4} with positive reach and m ∈ 𝕊^{3} such that

L^{1}({x · m : (x, m) ∈ nor X or (x, −m) ∈ nor X}) = 0  (4)

and the positive part of the mixed curvature measure C̄_{1,3}(X, m^{⊥}; ·) is infinite on a compact set.

Remark 1. (4) is the particular form of (1) for X and Y = m^{⊥} (cf. [5, Assumption (3.1)]).

Proof. Let m ∈ 𝕊^{3} be given and let Z ⊆ ℝ^{4} be a four-dimensional cube of edge length a > 0 and two facets F_{1}, F_{2} perpendicular to m. Let there be in F_{1} a disjoint family of three-dimensional balls C_{i} of radii r_{i} > 0, i = 1, 2, . . .. Note that we have ∑_{i} r_{i}^{3} < ∞. Then, for each i, a unit four-dimensional ball B_{i} is placed so that its centre lies outside of Z and B_{i} ∩ F_{1} = C_{i}. The compact set X = Z \ ⋃_{i}(int B_{i}) has reach greater than 3/4. Let a_{i} ≡ a_{i}(x, n) be the (generalized) principal directions and κ_{i} ≡ κ_{i}(x, n) the corresponding (generalized) principal curvatures of X at (x, n), i = 1, 2, 3. Using [5, Theorem 3.1] (or, equivalently, applying directly [6, Theorem 2]), one gets the formula

C̄_{1,3}(X, m^{⊥}; A × B)
= \frac{1}{4π} ∫_{nor X} 1_{A}(x)\frac{κ_{1}κ_{2}(a_{3} · m)^{2} + κ_{1}κ_{3}(a_{2} · m)^{2} + κ_{2}κ_{3}(a_{1} · m)^{2}}{| sin ∠(m, n)|^{3}\sqrt{1 + κ_{1}^{2}}\sqrt{1 + κ_{2}^{2}}\sqrt{1 + κ_{3}^{2}}} H^{3}(d(x, n))

valid for any bounded Borel subsets A ⊆ ℝ^{4} and B ⊆ m^{⊥} with L^{3}(B) = 1. If x ∈ A_{i} = int Z ∩∂B_{i} (A_{i} is the surface of the spherical hole digged by B_{i} in X) then κ_{1} = κ_{2} = κ_{3} = −1 and the principal directions can be chosen so that a_{1}, a_{2} ⊥ m and a_{3} · m = sin ∠(m, n). We thus obtain

C̄_{1,3}(X, m^{⊥}; A_{i} × B) = \frac{1}{8\sqrt{2}π} ∫_{nor X} 1_{A_{i}}(x)\frac{1}{| sin ∠(m, n)|}H^{3}(d(x, n)).

Consider the projection ψ : (x, n) ↦ m · n restricted to nor X ∩ (A_{i} × 𝕊^{3}), with Jacobian J_{1}ψ(x, n) = \frac{1}{\sqrt{2}} sin ∠(m, n). Applying the co-area formula, we get

C̄_{1,3}(X, m^{⊥}; A_{i} × B) = \frac{1}{8π} ∫_{√\overline{1−r_{i}^{2}}}^{1} \frac{1}{1 − u^{2}} H^{2}(ψ^{−1}u)du
= \frac{1}{8π} ∫_{√\overline{1−r_{i}^{2}}}^{1} \frac{1}{1 − u^{2}} 8π(1 − u^{2})du
= 1 − \sqrt{1 − r_{i}^{2}}.

Thus, choosing the r_{i}’s so that ∑_{i} r_{i}^{2} = ∞ (to see that this is possible, consider a partition of the cube Z into infinitely many parallel slices and fill a constant volume proportion of each slice by balls of diameters equal to the slice height), the positive part of C̄_{1,3}(X, m^{⊥}; X × B × 𝕊^{3}) will be greater or equal to

∑_{i} C̄_{1,3}^{+}(X, m^{⊥}; A_{i} × B) = ∑_{i} (1 − \sqrt{1 − r_{i}^{2}}) = ∞.

To see that (1) is satisfied, note that the set {x · m : (x, m) ∈ nor X or (x, −m) ∈ nor X} \ (F_{1} ∪ F_{2}) is countable. □

Remark 2. The construction from the proof can be easily adapted to ℝ^{5}, yielding a set X̃ with positive reach and m ∈ 𝕊^{4} with locally unbounded negative part of C̄_{1,4}(X̃, m^{⊥}; ·). Embedding then the set X from the example into a hyperplane in ℝ^{5} containing m, we get the positive part of C̄_{1,4}(X, m^{⊥}; ·) locally unbounded (note that the intersection of X with a translate of m^{⊥} is the same set, whether in ℝ^{4}, or embedded in ℝ^{5}, and its zeroth curvature measure - which is obviously connected with the mixed curvature measures in question - does not depend on the embedding space). Thus, the union of disjoint translates of X and X̃ presents an example of a set with positive reach in ℝ^{5} which has locally unbounded positive and negative parts of the mixed curvature measure with the hyperplane m^{⊥}.

Theorem 1. If for any compact subset K ⊆ ℝ^{4d}

∫_{K∩(nor X×nor Y)×R} ( sin ∠(m, n))^{3−d} H^{2d−2}(d(x, m, y, n)) < ∞,

then (2) holds. Consequently, (2) is satisfied automatically in dimensions d ≤ 3. Moreover, for any X, Y ⊆ ℝ^{d} with positive reach, (2) is satisfied by X, ρY for ϑ_{d}-almost all rotations ρ ∈ SO(d), where ϑ_{d} is the rotation invariant probability measure on SO(d).

Proof. Note that F_{r,s}(θ) ≤ (sin θ)^{−(2d−1−r−s)} for any θ ∈ [0, π] and

[⋀_{i∈I} a_{i}, ⋀_{j∈J} b_{j}] ≤ sin ∠(m, n)

for all I, J ⊆ {1, …, d − 1}. Thus, the first statement follows directly from (3). The last statement follows from the first one by integration over SO(d) since

∫_{SO(d)} ( sin ∠(m, ρn))^{3−d}ϑ_{d}(dρ)

is a finite constant depending on d only (see [7, Proposition 4.6] for a more detailed explanation). □

In the sequel, let G(d, k) denote the Grassmannian of k-dimensional linear subspaces in ℝ^{d} endowed with the rotational invariant probability measure ν_{k}^{d}. For X ⊆ ℝ^{d} with positive reach, 0 ≤ k ≤ d − 1 and W ∈ G(d, k + 1), we define the mapping

t_{W} : nor X ∩ (ℝ^{d} × W) → W × W, (x, m) ↦ (p_{W} x, m),

where p_{W} is the orthogonal projection from ℝ^{d} onto W. Further, let π_{W} : (z, m) ↦ z denote the first coordinate projection defined on the image of t_{W}. The image T_{W} X = (π_{W} ∘ t_{W})(nor X ∩ (ℝ^{d} × W)) will be called the tangential projection of X onto W. Note that T_{W} X is a closed subset of ℝ^{d} since nor X is a closed set. We need the following fact (cf. [8, p. 230]). Let R_{W} X denote the set of all points z ∈ T_{W} X such that (π_{W} ∘ t_{W})^{−1}{z} is either a singleton or a pair of points of the form (x, m), (x, −m) ∈ nor X.

Lemma 1. We have

ν_{k+1}^{d}({W ∈ G(d, k + 1) : H^{k}(T_{W} X \ R_{W} X) > 0}) = 0.

In other words, for ν_{k+1}^{d}-almost all W ∈ G(d, k + 1) and H^{k}-almost all z ∈ T_{W} X, if π_{W} (t_{W} (x, m)) = π_{W} (t_{W} (y, n)) = z then x = y and m = ±n.

Proof. Consider the countably p-rectifiable set

G = {(x, m, W) ∈ nor X × G(d, k + 1) : m ∈ W },
p = d − 1 + k(d − 1 − k) = k + (k + 1)(d − 1 − k)

(G corresponds to the (d − 1 − k)th Grassmann bundle of X introduced in [8]). The points (x, m) ∈ nor X where Tan(nor X, (x, m)) is not a (d − 1)-dimensional linear subspace have H^{d−1}-measure zero, thus, their contribution to the tangential projections can be neglected since the corresponding set G_{0} ⊆ G has H^{p}-measure zero. Let G_{1}, G_{2} be the set of such points (x, m, W) ∈ G where

dim Tan(nor X, (x, m)) ∩ (ℝ^{d} × W) > k,
dim Tan(nor X, (x, m)) ∩ ({0} × W) > 0,

respectively. The co-area formula applied to the projection F : (x, m, W) ↦ W defined on G yields

∫_{G(d,k+1)} H^{k}(G_{1} ∩ F^{−1}(W))ν_{k+1}^{d}(dW) = ∫_{G_{1}} J_{(k+1)(d−1−k)}F dH^{p}.

Since the Jacobian J_{(k+1)(d−1−k)}F vanishes on G_{1} (ker DF (x, m, W) contains all vectors (v, w, 0) such that (v, w) ∈ Tan(nor X, (x, m)) and w ∈ W and has thus dimension at least k + 1), the last integral vanishes and the contribution of points from G_{1} to the tangential projections can be neglected. Consider further the projection f̄ : (x, m, W) ↦ (p_{W} x, W) defined again on G. We have J_{p}f̄ = 0 on G_{2} and, consequently, H^{p}(f̄(G_{2})) = 0 by the area formula and the contribution of points from G_{2} to the tangential projections can be neglected as well.

Finally, we consider the tangential projection of points from G_{3} = G \ (G_{0} ∪ G_{1} ∪ G_{2}). Denote

Z = {(x, m, y, n, W) ∈ nor X × nor X × G(d, k + 1) :
(x, m, W) ∈ G_{3}, (y, n, W) ∈ G_{3}, 0 ≠ x − y ⊥ W }.

In fact, Z consists of pairs of ‘regular’ points from the unit normal bundle of X which have different first coordinates and whose first coordinates of the tangential projections coincide. (Note that we need not consider pairs with x = y and m, n linearly independent since both of these points fall into G_{2} with the corresponding subspace W.) We shall show that H^{p}(Z) = 0. Consequently, the image of Z under the Lipschitz mapping f̃: (x, m, y, n, W) ↦ (p_{W} x, W) has again H^{p}-measure zero and the assertion follows then when applying the co-area theorem for the projection on the second coordinate on the set f̃(Z).

Note that the points of Z are zero points of the Lipschitz mapping

h : (x, m, y, n, W) ↦ (W ∧ m, W ∧ n, W^{⊥} ∧ (x − y)).

Choose any 0 < ε < reach X and let Z_{ε} denote the image of Z under the bi-Lipschitz mapping

ϕ_{ε} : (x, m, y, n, W) ↦ (x + εm, y + εn, W).

Z_{ε} is a subset of the C^{1} manifold ∂X_{ε} × ∂X_{ε} × G(d, k + 1) of dimension

q = 2d − 2 + (k + 1)(d − 1 − k)

(here X_{ε} denotes the ε-parallel body to X) and the Lipschitz mapping h ∘ ϕ_{ε}^{−1} vanishes on Z_{ε}. Further, for any (x, m, y, n, W) ∈ Z, h ∘ ϕ_{ε}^{−1} is differentiable at z = ϕ_{ε}(x, m, y, n, W) and

rank D(h ∘ ϕ_{ε}^{−1})(z) ≥ 2d − 1 − k,

since im D(h ∘ ϕ_{ε}^{−1})(z) contains the vectors

\begin{matrix}(W ∧ v, 0, 0), & v ∈ W^{⊥}, \\ (0, W ∧ v, 0), & v ∈ W^{⊥}, \\ ((W ∩ u^{⊥}) ∧ (x − y) ∧ m, (W ∩ u^{⊥}) ∧ (x − y) ∧ n, W^{⊥} ∧ u), & u ∈ W. \end{matrix}

Using [1, Lemma 6.1], we infer that Z_{ε} is countably (p − 1)-rectifiable, since p − 1 = q − (2d − 1 − k). Hence, H^{p}(Z_{ε}) = 0 and, consequently, also H^{p}(Z) = 0, which completes the proof. □

Remark 3. Using the notation from the proof of Lemma 1, we have that

f̄(G) = {(z, W) : z ∈ T_{W} X, W ∈ G(d, k + 1)}

is countably p-rectifiable and H^{p}-mesurable. Consequently, applying [2, §3.2.22] for the projection (z, W) ↦ W, we get that T_{W} X is (H^{k}, k)-rectifiable and H^{k}-measurable for H^{(k+1)(d−1−k)}-almost all W ∈ G(d, k + 1) and the function W ↦ H^{k}(T_{W} X) is measurable.

The absolute curvature measure of X of order k can be defined by

∫ g(x, m)C_{k}^{abs}(X; d(x, m))
= c_{d,k} ∫_{G(d,k+1)} ∫ g ∘ t_{W}^{−1}(z, m)C_{k}(T_{W} X; d(z, m))ν_{k+1}^{d}(dW),  (5)

where the measure C_{k}(T_{W} X; ·) on T_{W} X × 𝕊^{k} is defined by

C_{k}(T_{W} X; ·) = ∫_{T_{W} X} H^{0}(· ∩ π_{W}^{−1}{z})H^{k}(dz)

and

c_{d,k} = \binom{d − 1}{k} \frac{H^{(k+1)(d−1−k)}(G(d, k + 1))}{H^{d−1−k}(𝕊^{d−1−k})H^{k(d−1−k)}(G(d − 1, k))}.

Note that t_{W}^{−1}(z, m) is correctly defined for almost all (z, m) by Lemma 1.

Remark 4. This definition is equivalent to the first equality in [8, Equation (3)]. The second equality in [8, Equation (3)] holds as well if X is d-dimensional in the sense that there is no (x, m) ∈ nor X with (x, −m) ∈ nor X. (For lower dimensional sets, the image points of the mapping f introduced in [8] have typically two pre-images.) [8, Equation (4)] holds in general since it has been derived from the first equality.

We shall say that a set X ⊆ ℝ^{d} with positive reach has locally bounded tangential projections if for any compact subset K ⊆ ℝ^{d} and 0 ≤ k ≤ d − 2,

sup_{W ∈G(d,k+1)} H^{k}(T_{W} X ∩ p_{W} K) < ∞.  (6)

Note that

C_{k−1}(T_{W} X, p_{W} K × 𝕊^{k}) = ∫_{T_{W} X∩p_{W} K} H^{0}(π_{W}^{−1}{z})H^{k}(dz)
≤ 2H^{k}(p_{W} K ∩ T_{W} X)

for almost all W ∈ G(d, k+1), since H^{0}(π_{W}^{−1}{z}) ≤ 2 for H^{k}-almost all z ∈ T_{W} X by Lemma 1.

We present now a sufficient condition for (2). As an immediate consequence one obtains that any two convex sets satisfy (2) (cf. [7, Proposition 4.5]).

Lemma 2. Let X, Y ⊆ ℝ^{d} have positive reach and assume that X, αY fulfil (1) for each 0 < α ≤ 1 and that for any 0 ≤ k ≤ d − 1 and bounded Borel subsets A, B ⊆ ℝ^{d},

sup_{α∈I} ∫_{A} C_{k}^{abs}(X ∩ (αY + z); B × 𝕊^{d−1})L^{d}(dz) < ∞

for some subset I ⊆ (0, ∞) of cardinality d + 1. Then X, Y satisfy (2).

Remark 5. Note that, due to the positive homogeneity of mixed curvature measures (see [6, Proposition 1]), if X, Y satisfy (2) then also X, αY satisfy (2) for any α > 0.

Proof. We shall assume without loss of generality that X, Y are compact. Then we have by the assumption

c(k) := sup_{α∈I} ∫_{ℝ^{d}} C_{k}^{abs}(X ∩ α(Y + z); ℝ^{d} × 𝕊^{d−1})L^{d}(dz) < ∞

for any 0 ≤ k ≤ d − 1. Since the absolute curvature measures majorize the total variations of the curvature measures (see [8]), we have for any measurable function h on ℝ^{2d} × 𝕊^{d−1} with |h| ≤ 1

| ∫ h(z, x, u) C_{k}(X ∩ (αY + z); d(x, u))L^{d}(dz)| ≤ c(k).

Assume, on the contrary, that (2) does not hold, and let s_{0} be the least natural number such that |C|_{r_{0},s_{0}}(X, Y; ·) is unbounded for some d − s_{0} ≤ r_{0} ≤ d − 1. Let A ⊆ ℝ^{2d} × 𝕊^{d−1} be a Borel set which supports an unbounded (positive or negative) part of |C|_{r_{0},s_{0}}(X, Y; ·). Given ε > 0, denote

A_{ε} = A ∩ f(((nor X × nor Y) ∩ R^{ε}) × [0, 1]),

where R^{ε} = {(x, m, y, n) ∈ (ℝ^{d})^{4} : ∠(m, n) ≤ π − ε}. Then all the mixed curvature measures are locally bounded on A^{ε} (cf. [7, Definition 4.1]) and the translative integral formula [6, Theorem 1] can be applied for the function h^{ε}(z, x, u) = 1_{A^{ε}}(x, x − z, u) and k = r_{0} + s_{0} − d, yielding

∫ h^{ε}(z, x, u) C_{k}(X ∩ (Y + z); d(x, u))L^{d}(dz) = ∑_{s=k}^{d+k} C_{d+k−s,s}(X, Y; A^{ε}).

Replacing Y with αY and taking into account the positive homogeneity of mixed curvature measures, we get

∫ h_{α}^{ε}(z, x, u) C_{k}(X ∩ (αY + z); d(x, u))L^{d}(dz) = ∑_{s=k}^{d+k} α^{s}C_{d+k−s,s}(X, Y; A^{ε})

for the function h_{α}^{ε}(z, x, u) = 1_{A^{ε}}(x, α^{−1}(x − z), u) with |h_{α}^{ε}| ≤ 1. The left hand side is bounded by c(k) for any ε > 0 and α ∈ I by the assumption. Since any polynomial in ℝ^{d} is uniquely determined by its values at any d + 1 different points, all the coefficients of the polynomial in α on the right hand side of the last equation have to be uniformly bounded in ε as well. This contradicts the fact that

lim_{ε→0_{+}} |C_{r_{0},s_{0}}(X, Y; A^{ε})| = |C_{r_{0},s_{0}}(X, Y; A)| = ∞.  □

Finally, we present a sufficient condition for a set with positive reach to satisfy (1) and (2) with any affine subspace of ℝ^{d}. Under this condition, the translative formulae from [5] can be applied.

Theorem 2. Let X ⊆ ℝ^{d} with positive reach have locally bounded tangential projections. Then, for any 0 < j ≤ d and L ∈ G(d, j), the pair X, L satisfies (1) and (2).

Proof. First, we shall verify (1). We have to show that the Lebesgue measure of

N = {z ∈ ℝ^{d} : ∃(x, m) ∈ nor X, m ⊥ L, p_{L^{⊥}}x = p_{L^{⊥}}z}

is zero. Note that N = T_{L^{⊥}}X ⊕ L. The H^{d−j−1}-measure of T_{L^{⊥}}X is locally bounded by assumption. Hence, the H^{d−1}-measure of N is locally bounded, which implies L^{d}(N) = 0. Note that (1) implies that X ∩ (L + z) has positive reach for L^{d−j}-almost all shifts z ∈ L^{⊥} (cf. [1, Theorem 4.10], [6, Proof of Theorem 1]).

To verify (2), we shall assume without loss of generality that X is compact. Due to Lemma 2, it suffices to show that for any 0 ≤ k ≤ j − 1,

∫ C_{k}^{abs}(X ∩ (L + z); ℝ^{d} × 𝕊^{d−1})L^{d−j}(dz) < ∞.

Due to the considerations above, it is enough to integrate over the complement of T_{L^{⊥}}X in L^{⊥}, hence we can neglect the points (x, m) ∈ nor X with m ⊥ L. Note that the total (absolute) curvature measures are independent of the dimension of the embedding space (cf. [8, Theorem 2]). Hence, we can treat the sets X ∩ (L + z) as subsets of the j-dimensional space L and we have by definition

C_{k}^{abs}(X ∩ (L + z); L × 𝕊^{j−1})
≤ 2c_{j,k} ∫_{G(j,k+1)} H^{k}(T_{W} (X ∩ (L + z)))ν_{k+1}^{j}(dW)

for L^{d−j}-almost all z ∈ L^{⊥} (for the existence of the last integral, see Remark 3). Let W ∈ G(j, k + 1) be fixed and consider the subspace W̃ = W + L^{⊥} ∈ G(d, d + k − j + 1). For any z ∈ L^{⊥} \ T_{L^{⊥}}X we have

T_{W} (X ∩ (L + z)) = T_{W} X ∩ p_{L^{⊥}}^{−1}{z}.

Applying [4, §7.7] or [2, §2.10.25] for the set T_{W} X and projection z ↦ p_{L^{⊥}}z, we get

∫_{L^{⊥}} H^{k}(T_{W} (X ∩ (L + z)))L^{d−j}(dz) ≤ const · H^{d+k−j}(T_{W} X),

where the last term is bounded uniformly in W by the assumption and the assertion follows. □

Remark 6. Theorems 1, 2 hold for locally finite unions of sets with positive reach (see [7]) as well. For these sets, an additional multiplicity term i_{X}(x, m)i_{Y} (y, n) appears under the integral in (3), but both index functions i_{X}, i_{Y} are locally uniformly bounded.

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Received July 18, 2000; revised version November 13, 2000