Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 40 (1999), No. 1, 113-124.

Kinematic Formulae for Support Measures of Convex Bodies

Stefan Glasauer^{∗}

Mathematisches Institut, Universität Freiburg Eckerstr. 1, D-79104 Freiburg, Germany e-mail: glasauer@mathematik.uni-freiburg.de

Abstract. A new kinematic formula for the support measures of convex bodies is proved. The underlying operation is the convex hull operation for pairs of convex bodies. Further, the concept of mixed support measures is introduced and kinematic relations for these new functionals are indicated. MSC 1991: 52A20, 52A22 Keywords: convex body, convex hull operation, support measure, principal kinematic formula.

1. Introduction

The main purpose of this paper is to establish a new kinematic formula for support measures (or generalized curvature measures) of convex bodies. The proof depends on a recent result of Schneider [11], which confirmed a conjecture of the author [4, 6]. For surveys on integral geometry of convex bodies, the reader is referred to Schneider and Wieacker [13], Schneider and Weil [12], and Section 4.5 in Schneider [9]. The latter book is the basic reference for the geometric notions used below. For the measure theoretic results we will need, see Ash [1] and Cohn [2], for example.

Let K be the set of all convex bodies, i.e. the set of all non-empty, compact, convex subsets of R^{n}, and let K be endowed with the Hausdorff metric. Let ⟨·, ·⟩ be the Euclidean inner product of R^{n}. Let B^{n} be the unit ball in R^{n} and S^{n−1} its boundary. For K ∈ K, let NorK be the set of all support elements of K, i.e. the set of all pairs (x, u) ∈ Σ := R^{n} × S^{n−1} where x is a boundary point of K and u is an outer unit normal vector of K at x. The support measures (or generalized curvature measures) of a convex body K are common generalizations of Federer’s curvature measures and the area measures of Aleksandrov-Fenchel-Jessen. They are the unique Borel measures Θ_{0}(K; ·), …, Θ_{n−1}(K; ·) on Σ which are concentrated on NorK and satisfy the Steiner-type relation

∗The work on this paper was supported in part by the Deutsche Forschungsgemeinschaft.

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 1999 Heldermann Verlag

∫_{R^{n}\K} f dλ^{n} = ∑_{j=0}^{n−1} \binom{n − 1}{j} ∫_{0}^{∞}∫_{Σ} t^{n−j−1}f(x + tu) dΘ_{j}(K; (x, u)) dt  (1)

for all Lebesgue integrable f : R^{n} → R; here λ^{n} is the Lebesgue measure on R^{n}. We will make essential use of the fact that the maps K ↦ Θ_{j}(K; ·) are weakly continuous, i.e. if lim K_{i} = K, then lim ∫ f dΘ_{j}(K_{i}; ·) = ∫ f dΘ_{j}(K; ·) for all bounded Borel functions f : Σ → R that are continuous Θ_{j}(K; ·)-almost everywhere.

We denote by B(X) the Borel σ-algebra of a topological space X. The measures defined by C_{j}(K; A) := Θ_{j}(K; A × S^{n−1}), A ∈ B(R^{n}), are, apart from normalizing constants, Federer’s curvature measures, and the equation S_{j}(K; ω) := Θ_{j}(K; R^{n} × ω), ω ∈ B(S^{n−1}), defines the area measures of Alexandrov-Fenchel-Jessen. The global measure Θ_{j}(K; Σ) equals, up to constant multipliers, the quermassintegral W_{n−j}(K) and the intrinsic volume V_{j}(K).

The Haar measure on the unimodular group G_{n} of all proper rigid motions of R^{n} is denoted by μ, where the normalization is chosen so that μ({g ∈ G_{n} : g(0) ∈ B^{n}}) equals κ_{n}, the volume of the unit ball B^{n}. For η, η′ ⊂ Σ and a motion g ∈ G_{n}, let gη := {(gx, g_{0}u) ∈ Σ : (x, u) ∈ η}, where g_{0} is the rotational part of g, and

η ∧ η′ := {(x, u) ∈ Σ : there are u_{1}, u_{2} ∈ S^{n−1} with (x, u_{1}) ∈ η,
(x, u_{2}) ∈ η′, and u ∈ pos{u_{1}, u_{2}}} ,

where pos denotes the positive hull operation.

The following theorem can be viewed as an extension of a special case of Federer’s principal kinematic formula [3]. In Theorems 1 and 3 below, we assume that the domains of the measures Θ_{j}(K; ·) are extended to the respective completions of B(Σ).

Theorem 1. Let K, K′ ∈ K, η, η′ ∈ B(Σ), and j ∈ {0, …, n − 2}. Then we have

∫_{{g∈G_{n}:K∩gK′≠∅}} Θ_{j}(K ∩ gK′; (η ∩ NorK) ∧ g(η′ ∩ NorK′)) dμ(g)

= \frac{κ_{n−j}}{nκ_{n}κ_{j}} ∑_{k=j+1}^{n−1} \binom{n − j}{k − j}\frac{κ_{k}κ_{n+j−k}}{κ_{n−k}κ_{k−j}} Θ_{k}(K; η)Θ_{n+j−k}(K′; η′) .

Theorem 1 was proved in Glasauer [5] under some restrictions on the bodies K, K′ (see [5], Theorem 3.1), which can now be removed by the following recent result of Schneider [11]. Recall that the normal cone N(K, x) of a convex body K at a boundary point x is the set of all outer normal vectors of K at x. The boundary of K ∈ K is denoted by bdK.

Theorem 2. (Schneider) Let K, K′ ∈ K. Then for μ-almost all g ∈ G_{n}, the linear hulls of the normal cones N(K, x), N(gK′, x) have trivial intersection for all x ∈ bdK ∩ bdgK′.

Remark. There is an extension of Theorem 1 to finite unions of convex bodies, see [5], Theorem 3.3. It is likely that there are extensions to more general sets, but this remains an open problem.

The main purpose of this paper is to prove a “dual” counterpart to Theorem 1, which can be stated as follows. For η, η′ ⊂ Σ let

η ∨ η′ := {(x, u) ∈ Σ : there are x_{1}, x_{2} ∈ R^{n} with x_{1} − x_{2} ⊥ u, (x_{1}, u) ∈ η,
(x_{2}, u) ∈ η′, and x ∈ [x_{1}, x_{2}]} ,

where [x_{1}, x_{2}] is the closed line segment with endpoints x_{1}, x_{2}. The convex hull of the union of two convex bodies K, K′ is denoted by K ∨ K′.

Theorem 3. Let j ∈ {1, …, n − 1}. Then we have

lim_{r→∞} \frac{1}{r^{n+1}} ∫_{\{g∈G_{n}:gK′⊂rB^{n}\}} Θ_{j}(K ∨ gK′; (η ∩ NorK) ∨ g(η′ ∩ NorK′)) dμ(g)

= \frac{κ_{n−1}}{(n + 1)nκ_{n}} ∑_{k=0}^{j−1} Θ_{k}(K; η)Θ_{j−k−1}(K′; η′)

uniformly for all η, η′ ∈ B(Σ) and all K, K′ ∈ K contained in a fixed ball.

The proof of Theorem 3 will be given in the next section. An essential ingredient is the following recent result of Schneider [11] (the formulation there is equivalent to the one given here).

Theorem 4. (Schneider) Let K, K′ ∈ K. Then for μ-almost all g ∈ G_{n}, the following is true. For each point x ∈ (bd(K ∨ gK′))\(K ∪ gK′), there are unique points y ∈ K and z ∈ gK′ with x ∈ [y, z].

Both Theorem 3 and Theorem 4 were conjectured by the author in [6], where it was stated that Theorem 3 can be proved if Theorem 4 holds true. Special cases of Theorems 2 and 4 were proved in [4, 5, 6]. Counterparts to Theorems 1–4 in spherical space were established in [4].

The setup of the paper is as follows. Section 2 contains the proof of Theorem 3. In Section 3, the concept of mixed support measures is introduced and generalizations of Theorems 1 and 3 for these new functionals are stated.

2. Proof of Theorem 3

Let us first observe that for strictly convex K, K′, Theorem 3 follows from a known result.

Lemma 1. Theorem 3 holds if K and K′ are strictly convex.

Proof. The following is a special case of Theorem 5 in [6]: Let j ∈ {0, …, n − 1}. Then

lim_{r→∞} \frac{1}{r^{n+1}} ∫_{\{g∈G_{n}:gK′⊂rB^{n}\}} S_{j}(K ∨ gK′; ω ∩ g_{0}ω′) dμ(g)

= \frac{κ_{n−1}}{(n + 1)nκ_{n}} ∑_{k=0}^{j−1} S_{k}(K; ω)S_{j−k−1}(K′; ω′)  (2)

uniformly for all ω, ω′ ∈ B(S^{n−1}) and all K, K′ ∈ K which are contained in a fixed ball.

For strictly convex K ∈ K and η ∈ B(Σ) we define ω_{K,η} := {u ∈ S^{n−1} : (x, u) ∈ η ∩ NorK for an x ∈ R^{n}}. Let h_{K} : S^{n−1} → R be the support function of K, i.e. h_{K}(u) := max_{x∈K}⟨x, u⟩ = ⟨τ_{K}(u), u⟩ for u ∈ S^{n−1}, where τ_{K}(u) is the boundary point of K with outer normal vector u.

Now let K, K′ ∈ K be strictly convex and let η ⊂ NorK and η′ ⊂ NorK′ be Borel sets. We have

η ∨ η′ = (R^{n} × (ω_{K,η} ∩ ω_{K′,η′} ∩ {h_{K} = h_{K′}})) ∩ Nor(K ∨ K′) .

In fact, let (x, u) ∈ η ∨ η′. Let x_{1}, x_{2} ∈ R^{n} with ⟨x_{1}, u⟩ = ⟨x_{2}, u⟩, (x_{1}, u) ∈ η, (x_{2}, u) ∈ η′, and x ∈ [x_{1}, x_{2}]. We have h_{K}(u) = h_{K′}(u) = ⟨x_{1}, u⟩ = ⟨x, u⟩. Thus (x, u) ∈ Nor(K ∨ K′) and u ∈ ω_{K,η} ∩ ω_{K′,η′} ∩ {h_{K} = h_{K′}}. Now let (x, u) ∈ Nor(K ∨ K′) with u ∈ ω_{K,η} ∩ ω_{K′,η′} and h_{K}(u) = h_{K′}(u). Let x_{1} := τ_{K}(u), x_{2} := τ_{K′}(u). It follows that (x_{1}, u) ∈ η, (x_{2}, u) ∈ η′, and x ∈ [x_{1}, x_{2}], as required.

We infer from this equation that

S_{j}(K ∨ K′; ω_{K,η} ∩ ω_{K′,η′}) − S_{j}(K; S^{n−1}) − S_{j}(K′; S^{n−1})
≤ S_{j}(K ∨ K′; ω_{K,η} ∩ ω_{K′,η′} ∩ {h_{K} = h_{K′}})
= Θ_{j}(K ∨ K′; η ∨ η′)
≤ S_{j}(K ∨ K′; ω_{K,η} ∩ ω_{K′,η′})

for all j ∈ {0, …, n − 1}. We also have

Θ_{j}(K; η) = S_{j}(K; ω_{K,η}), Θ_{j}(K′; η′) = S_{j}(K′; ω_{K′,η′}) .

Now the assertion follows from equation (2). □

We now want to extend our formula to general convex bodies by approximation.

We collect some notations which will be used below. Let K, K′ ∈ K. We define Σ_{1}(K, K′) as the set of all (x, u) ∈ NorK ∪ NorK′ such that there is a hyperplane which supports both K and K′, with both bodies on the same side of it, and which is orthogonal to u and contains x. We let Σ_{2}(K, K′) := Nor(K ∨ K′)\(NorK ∪ NorK′). We define G(K, K′) to be the set of all motions g ∈ G_{n} such that for each x ∈ (bd(K ∨ gK′))\(K ∪ gK′) there are unique points y ∈ K, z ∈ gK′ with x ∈ [y, z]. Theorem 4 says that for all K, K′ ∈ K there is a Borel subset of G(K, K′) whose complement has μ-measure zero. For g ∈ G(K, K′) we define two maps, π_{g,K,K′} and π′_{g,K,K′}, on Σ_{2}(K, gK′) by the requirement that π_{g,K,K′}(x, u) := (y, u), π′_{g,K,K′}(x, u) := (g^{−1}z, g_{0}^{−1}u), where y, z are the unique points in K, gK′, respectively, with x ∈ [y, z]. The images of π_{g,K,K′}, π′_{g,K,K′} are contained in NorK, NorK′, respectively.

Lemma 2. For μ-almost all g ∈ G_{n}, we have Θ_{j}(K ∨ gK′; Σ_{1}(K, gK′)) = 0 for all j ∈ {0, …, n − 1}.

Proof. Let j ∈ {0, …, n − 1}. Let E be the set of all hyperplanes in R^{n}, and let E(K) be the subset of all support planes of K (these sets are endowed with their natural topologies). For A ⊂ E, let η_{A} := {(x, u) ∈ Σ : x ∈ H, H ⊥ u for an H ∈ A}. Define a measure ρ(K; ·) on E by

ρ(K; A) := Θ_{j}(K; η_{A}), A ∈ B(E).

We have Σ_{1}(K, gK′) ⊂ η_{E(K)∩E(gK′)} ∩ (NorK ∪ NorgK′) and therefore

Θ_{j}(K ∨ gK′; Σ_{1}(K, gK′)) ≤ ρ(K; E(K) ∩ E(gK′)) + ρ(gK′; E(K) ∩ E(gK′)) .

We will show that both terms in this sum are zero for μ-almost all g. We denote the rotation invariant probability measure on the rotation group SO_{n} by ν. The line through 0 orthogonal to a hyperplane H is written as H^{⊥}. The Fubini theorem shows that

∫_{G_{n}} ρ(K; E(K) ∩ E(gK′)) dμ(g)

= ∫_{SO_{n}} ∫_{R^{n}} ∫_{E} 1_{E(K)}(H)1_{E(ϑK′)+x}(H) dρ(K; H)dλ^{n}(x)dν(ϑ)
= ∫_{SO_{n}} ∫_{E} 1_{E(K)}(H) ∫_{H} ∫_{H^{⊥}} 1_{E(ϑK′)}(H + y + z) dλ^{1}(y)dλ^{n−1}(z)dρ(K; H)dν(ϑ)
= 0 ,

since H + y ∈ E(ϑK′) for at most two values of y ∈ H^{⊥}. Hence ρ(K; E(K) ∩ E(gK′)) = 0 for μ-almost all g, and the same reasoning applies to the second term. □

Lemma 3. Let j ∈ {0, …, n − 1}. Then we have

lim_{r→∞} \frac{1}{r^{n+1}} ∫_{\{g∈G_{n}:gK′⊂rB^{n}\}} ∫_{Σ_{2}(K,gK′)} (f ∘ π_{g,K,K′})(f′ ∘ π′_{g,K,K′}) dΘ_{j}(K ∨ gK′; ·)dμ(g)

= \frac{κ_{n−1}}{(n + 1)nκ_{n}} ∑_{k=0}^{j−1} ∫_{Σ} f dΘ_{k}(K; ·) ∫_{Σ} f′ dΘ_{j−k−1}(K′; ·)

uniformly for all continuous f, f′ : Σ → R with 0 ≤ f, f′ ≤ 1 and all strictly convex bodies K, K′ ∈ K contained in a fixed ball.

Proof. Let j ∈ {0, …, n − 1}, let K, K′ ∈ K be strictly convex, and let η, η′ ∈ B(Σ). We have G(K, K′) = G_{n}. By using Lemma 2 we obtain

Θ_{j}(K ∨ gK′; (η ∩ NorK) ∨ g(η′ ∩ NorK′))
= Θ_{j}(K ∨ gK′; ((η ∩ NorK) ∨ g(η′ ∩ NorK′))\Σ_{1}(K, gK′))
= Θ_{j}(K ∨ gK′; ((η ∩ NorK) ∨ g(η′ ∩ NorK′)) ∩ Σ_{2}(K, gK′))

= ∫_{Σ_{2}(K,gK′)} 1_{(η∩NorK)∨g(η′∩NorK′)} dΘ_{j}(K ∨ gK′; ·)

= ∫_{Σ_{2}(K,gK′)} (1_{η} ∘ π_{g,K,K′})(1_{η′} ∘ π′_{g,K,K′}) dΘ_{j}(K ∨ gK′; ·)

for μ-almost all g ∈ G_{n}. From this and Lemma 1 it follows that

lim_{r→∞} \frac{1}{r^{n+1}} ∫_{\{g∈G_{n}:gK′⊂rB^{n}\}} ∫_{Σ_{2}(K,gK′)} (f ∘ π_{g,K,K′})(f′ ∘ π′_{g,K,K′}) dΘ_{j}(K ∨ gK′; ·)dμ(g)

= \frac{κ_{n−1}}{(n + 1)nκ_{n}} ∑_{k=0}^{j−1} ∫_{Σ} f dΘ_{k}(K; ·) ∫_{Σ} f′ dΘ_{j−k−1}(K′; ·)

uniformly for all strictly convex K, K′ ∈ K contained in a fixed ball and all simple functions f, f′ : Σ → R with 0 ≤ f, f′ ≤ 1, i.e. functions of the type ∑_{i=1}^{m} c_{i}1_{η_{i}} with m ∈ N, 0 ≤ c_{i} ≤ 1, and pairwise disjoint Borel sets η_{1}, …, η_{m} ⊂ Σ. Since every non-negative continuous function is the pointwise limit of an increasing sequence of simple functions, the monotone convergence theorem shows the assertion. □

Lemma 4. Lemma 3 holds also for general convex bodies K, K′ ∈ K.

Proof. Let K, K′ ∈ K. Let (K_{i})_{i∈N}, (K′_{i})_{i∈N} be sequences of strictly convex bodies such that K_{i+1} ⊂ intK_{i}, K′_{i+1} ⊂ intK′_{i} for all i, where int denotes the interior, and lim K_{i} = K, lim K′_{i} = K′. (The existence of such sequences follows easily from Theorem 3.3.1 in Schneider [9].) Let g ∈ G(K, K′) be such that Θ_{j}(K ∨ gK′; Σ_{1}(K, gK′)) = 0. Since K_{i}, K′_{i} are strictly convex, we have G(K_{i}, K′_{i}) = G_{n} for i ∈ N. Let K_{0} := K, K′_{0} := K′. Let f, f′ ∈ C(Σ). Let m ∈ N. We define

A_{m} := \{(x, u) ∈ ⋃_{i∈N_{0}} Nor(K_{i} ∨ gK′_{i}) : |x − y| ≥ \frac{1}{m} for all (y, v) ∈ ⋃_{i∈N_{0}} Σ_{1}(K_{i}, gK′_{i})\},
B_{m} := \{(x, u) ∈ Σ : |x − y| ≥ \frac{1}{m} for all (y, v) ∈ ⋃_{i∈N_{0}} Σ_{1}(K_{i}, gK′_{i})\}.

The set A_{m} is a closed subset of B_{m}, since ⋃_{i∈N_{0}} Nor(K_{i} ∨ gK′_{i}) is closed, and also B_{m} is a closed set, hence a normal subspace of Σ.

Define a function g_{1} : A_{1} → R as follows. If (x, u) ∈ A_{1} ∩ Σ_{2}(K_{i}, gK′_{i}) for one (and only one) i ∈ N_{0}, let g_{1}(x, u) := f(π_{g,K_{i},K′_{i}}(x, u))f′(π′_{g,K_{i},K′_{i}}(x, u)). For the remaining (x, u) ∈ A_{1}, we define g_{1}(x, u) := 0.

The function g_{1} : A_{1} → R thus defined is continuous: Let (x, u) ∈ A_{1}, and let ((x_{m}, u_{m}))_{m∈N} be a sequence in A_{1} converging to (x, u). Then (x, u) ∈ Nor(K_{i} ∨ gK′_{i}) for a unique i ∈ N_{0}. The pair (x, u) is either in NorK_{i}, in NorgK′_{i}, or in Σ_{2}(K_{i}, gK′_{i}). If (x, u) ∈ NorK_{i}, then (x_{m}, u_{m}) ∈ ⋃_{i∈N_{0}} NorK_{i} for almost all m ∈ N and therefore g_{1}(x, u) = 0 and g_{1}(x_{m}, u_{m}) = 0 for almost all m. The case (x, u) ∈ NorgK′_{i} is analogous. If (x, u) ∈ Σ_{2}(K_{i}, gK′_{i}), then there is a sequence (i_{m})_{m∈N} of non-negative integers such that (x_{m}, u_{m}) ∈ Σ_{2}(K_{i_{m}}, gK′_{i_{m}}) for almost all m. Let y_{m} and z_{m} be the unique points in K_{i_{m}}, gK′_{i_{m}}, respectively, with x_{m} ∈ [y_{m}, z_{m}] and let y, z be the points in K_{i}, gK′_{i}, respectively, with x ∈ [y, z]. Assume that y_{m} does not converge to y for m → ∞. Then there is a convergent subsequence (y_{m_{k}})_{k∈N} with ȳ := lim_{k→∞} y_{m_{k}} ≠ y. Let z̄ be a limit point of the sequence (z_{m_{k}})_{k∈N}. We conclude that x ∈ [ȳ, z̄]. Since ȳ ∈ K_{i} and z̄ ∈ gK′_{i}, this contradicts the fact that g ∈ G(K_{i}, gK′_{i}). Thus lim y_{m} = y, and analogously we get lim z_{m} = z. Hence lim π_{g,K_{im},K′_{im}}(x_{m}, u_{m}) = π_{g,K_{i},K′_{i}}(x, u) and lim π′_{g,K_{im},K′_{im}}(x_{m}, u_{m}) = π′_{g,K_{i},K′_{i}}(x, u), and therefore lim g_{1}(x_{m}, u_{m}) = g_{1}(x, u), as required.

We can now apply Tietze’s extension theorem to extend g_{1} to a continuous function h_{1} on B_{1}. If m ∈ N and h_{m} : B_{m} → R is already defined, let g_{m+1} be the extension of h_{m} to the set B_{m} ∪ A_{m+1} which is defined in the same way as g_{1}. Then apply Tietze’s theorem to extend g_{m+1} to a continuous function h_{m+1} on B_{m+1}. In this way, we obtain continuous functions h_{m} : B_{m} → R with h_{m+1}|B_{m} = h_{m} for all m ∈ N.

Now define a function h : Σ → R as follows. If (x, u) ∈ Σ\ ⋃_{i∈N_{0}} Σ_{1}(K_{i}, gK′_{i}), then (x, u) ∈ B_{m} for a sufficiently large m, since ⋃_{i∈N_{0}} Σ_{1}(K_{i}, gK′_{i}) is closed; we let h(x, u) := h_{m}(x, u). If (x, u) ∈ ⋃_{i∈N_{0}} Σ_{1}(K_{i}, gK′_{i}), we let h(x, u) := 0. So if (x, u) ∈ Nor(K_{i} ∨ gK′_{i}) for an i ∈ N_{0}, then h(x, u) = f(π_{g,K_{i},K′_{i}}(x, u))f′(π′_{g,K_{i},K′_{i}}(x, u)) in the case (x, u) ∈ Σ_{2}(K_{i}, gK′_{i}) and h(x, u) = 0 if (x, u) ∈ NorK_{i} ∪NorgK′_{i}. Since Θ_{j}(K ∨gK′; Σ_{1}(K, gK′)) = 0, the function h is continuous Θ_{j}(K ∨ gK′; ·)-almost everywhere, and it is Borel measurable and bounded. Now the weak continuity of the support measures implies that

lim_{i→∞} ∫_{Σ_{2}(K_{i},gK′_{i})} (f ∘ π_{g,K_{i},K′_{i}})(f′ ∘ π′_{g,K_{i},K′_{i}}) dΘ_{j}(K_{i} ∨ gK′_{i}; ·)

= lim_{i→∞} ∫_{Σ} h dΘ_{j}(K_{i} ∨ gK′_{i}; ·)

= ∫_{Σ} h dΘ_{j}(K ∨ gK′; ·)

= ∫_{Σ_{2}(K,gK′)} (f ∘ π_{g,K,K′})(f′ ∘ π′_{g,K,K′}) dΘ_{j}(K ∨ gK′; ·).

The dominated convergence theorem, Lemma 2, and Theorem 4 show that

lim_{i→∞}  ∫_{{g∈G_{n}:gK′_{i}⊂rB^{n}}} ∫_{Σ_{2}(K_{i},gK′_{i})} (f ∘ π_{g,K_{i},K′_{i}})(f′ ∘ π′_{g,K_{i},K′_{i}}) dΘ_{j}(K_{i} ∨ gK′_{i}; ·)dμ(g)

=  ∫_{{g∈G_{n}:gK′⊂rB^{n}}} ∫_{Σ_{2}(K,gK′)} (f ∘ π_{g,K,K′})(f′ ∘ π′_{g,K,K′}) dΘ_{j}(K ∨ gK′; ·)dμ(g)  (3)

for all r > 0. According to Lemma 3, for a given ϵ > 0, there is an r_{0} > 0 such that for all r ≥ r_{0} we have

|\frac{1}{r^{n+1}} ∫_{\{g∈G_{n}:gK′_{i}⊂rB^{n}\}} ∫_{Σ_{2}(K_{i},gK′_{i})} (f ∘ π_{g,K_{i},K′_{i}})(f′ ∘ π′_{g,K_{i},K′_{i}}) dΘ_{j}(K_{i} ∨ gK′_{i}; ·)dμ(g)

− \frac{κ_{n−1}}{(n + 1)nκ_{n}} ∑_{k=0}^{j−1} ∫_{Σ} f dΘ_{k}(K_{i}; ·) ∫_{Σ} f′ dΘ_{j−k−1}(K′_{i}; ·)| ≤ ϵ

for all continuous f, f′ : Σ → R with 0 ≤ f, f′ ≤ 1 and all i ∈ N. In addition, r_{0} depends only on the radius of the smallest centered ball containing all K_{i}, K′_{i} and not on the particular choices of the bodies K_{i}, K′_{i}. Now (3) shows that

|\frac{1}{r^{n+1}} ∫_{\{g∈G_{n}:gK′⊂rB^{n}\}} ∫_{Σ_{2}(K,gK′)} (f ∘ π_{g,K,K′})(f′ ∘ π′_{g,K,K′}) dΘ_{j}(K ∨ gK′; ·)dμ(g)

− \frac{κ_{n−1}}{(n + 1)nκ_{n}} ∑_{k=0}^{j−1} ∫_{Σ} f dΘ_{k}(K; ·) ∫_{Σ} f′ dΘ_{j−k−1}(K′; ·)| ≤ ϵ

for all r ≥ r_{0}, and the proof of Lemma 4 is complete. □

Proof of Theorem 3. Let K, K′ ∈ K, and let η ∈ B(NorK), η′ ∈ B(NorK′). It can easily be shown that (η ∨ gη′)\Σ_{1}(K, gK′) is a Borel set for all g ∈ G(K, K′). Hence the map F_{η,η′} : g ↦ Θ_{j}(K ∨ gK′; η ∨ gη′) is defined on a set of full μ-measure. We want to show that F_{η,η′} is measurable. Assume that η and η′ are compact. Then there are decreasing sequences (f_{i})_{i∈N}, (f′_{i})_{i∈N} in C(Σ) with 0 ≤ f_{i}, f′_{i} ≤ 1 and lim f_{i} = 1_{η}, lim f′_{i} = 1_{η′} in the sense of pointwise convergence. The limit lim_{i→∞} 1_{Σ_{2}(K,gK′)}(f_{i} ∘ π_{g,K,K′})(f′_{i} ∘ π′_{g,K,K′}) equals the indicator function of the set (η∨gη′)\Σ_{1}(K, K′) for all g ∈ G(K, K′). Thus the monotone convergence theorem, Lemma 2, Theorem 4, and the proof of Lemma 4 show that F_{η,η′} is the limit of measurable functions μ-almost everywhere and hence is measurable on a set of full measure. Let η′ ⊂ NorK′ still be compact. The set of all η ∈ B(NorK) such that F_{η,η′} is measurable on a set of full measure can easily be shown to be a Dynkin system. Since it contains all compact sets, it must coincide with B(NorK), so F_{η,η′} has the required property for compact η′. We can now apply the same argument to see that F_{η,η′} is measurable on a set of full measure for all η ∈ B(NorK), η′ ∈ B(NorK′).

Let us define

φ(r, K, K′, η, η′) := \frac{1}{r^{n+1}} ∫_{\{g∈G_{n}:gK′⊂rB^{n}\}}Θ_{j}(K ∨ gK′; (η ∩ NorK) ∨ g(η′ ∩ NorK′)) dμ(g)

− \frac{κ_{n−1}}{(n + 1)nκ_{n}} ∑_{k=0}^{j−1} Θ_{k}(K; η)Θ_{j−k−1}(K′; η′)

for all r > 0, K, K′ ∈ K, and η, η′ ∈ B(Σ). It follows from Lemma 2, Theorem 4, and the monotone convergence theorem that φ(r, K, K′, η, η′) is a finite signed measure in both η and η′. Approximating the indicator functions of compact sets η, η′ ⊂ Σ by monotone sequences of continuous functions as before, we deduce from Lemma 4 that for all ϵ > 0 there is an r_{0} > 0 with

|φ(r, K, K′, η, η′)| ≤ ϵ

for all r ≥ r_{0}, all compact η, η′ ⊂ Σ, and all K, K′ ∈ K that are contained in a fixed ball. Using the regularity of finite signed measures on Σ, we deduce that this inequality also holds for Borel sets η and compact sets η′. In the same way we finally obtain it for arbitrary η, η′ ∈ B(Σ). This concludes the proof of Theorem 3. □

3. Mixed support measures and extensions of Theorems 1 and 3

In this section, we want to introduce the concept of mixed support measures and state generalizations of Theorems 1 and 3 for these new functionals.

Let S_{n−1}(K; ·) = Θ_{n−1}(K; R^{n} × ·) be the area measure of K ∈ K. For n-dimensional convex bodies K and Borel subsets ω of the sphere, S_{n−1}(K; ω) equals the (n−1)-dimensional Hausdorff measure of the set of all x ∈ bdK such that there exists an outer unit normal vector of K at x that belongs to ω. The mixed area measures S(K_{1}, …, K_{n−1}; ·) appear as coefficients in the polynomial expansion of the area measure of a Minkowski linear combination of convex bodies: we have

S_{n−1}(λ_{1}K_{1} + … + λ_{m}K_{m}; ·) = ∑_{i_{1},...,i_{n−1}=1}^{m} λ_{i_{1}} ⋯ λ_{i_{n−1}}S(K_{i_{1}}, …, K_{i_{n−1}}; ·)  (4)

for all m ∈ N, K_{1}, …, K_{m} ∈ K, λ_{1}, …, λ_{m} ≥ 0, see Schneider [9], p. 275.

Let S be the set of all strictly convex bodies. For M ∈ S and u ∈ S^{n−1}, we defined τ_{M}(u) to be the boundary point of M with outer normal vector u. Let s_{M} : S^{n−1} → Σ, u ↦ (τ_{M}(u), u). For a positive real number λ, we define a map t_{M}^{λ} : Σ → Σ by means of t_{M}^{λ}(x, u) := ((x − τ_{M}(u))/λ, u). Both s_{M} and t_{M}^{λ} are continuous and hence Borel measurable.

Theorem 5. For j ∈ {0, …, n−2}, K ∈ K, and M_{1}, …, M_{n−j−1} ∈ S, there are unique Borel measures Θ_{j}(K; M_{1}, …, M_{n−j−1}; ·) on Σ, such that they are symmetric in M_{1}, …, M_{n−j−1} and satisfy

t_{λ_{1}M_{1}+...+λ_{m}M_{m}}^{λ}(Θ_{n−1}(λK + λ_{1}M_{1} + … + λ_{m}M_{m}; ·)) = λ^{n−1}Θ_{n−1}(K; ·) +
+ ∑_{j=0}^{n−2} \binom{n − 1}{j}λ^{j} ∑_{i_{1},...,i_{n−j−1}=1}^{m} λ_{i_{1}} ⋯ λ_{i_{n−j−1}}Θ_{j}(K; M_{i_{1}}, …, M_{i_{n−j−1}}; ·)  (5)

for all m ∈ N, M_{1}, …, M_{m} ∈ S, λ > 0, and λ_{1}, …, λ_{m} ≥ 0. These measures are concentrated on NorK, and they depend weakly continuously on K ∈ K.

We call the functional Θ_{j}(K; ·; ·) the j-th mixed support measure of K. It can easily be deduced from (1) and (5) that Θ_{j}(K; B^{n}, …, B^{n}; ·) = Θ_{j}(K; ·) for all K ∈ K and j ∈ {0, …, n − 2}.

Proof. The uniqueness statement being clear, we need only prove existence. Let j ∈ {0, …, n−2}. From (4) we can infer that there are ℓ ∈ N, a_{i} ∈ R, λ^{(i)} > 0, λ_{1}^{(i)}, …, λ_{n−j−1}^{(i)} ≥ 0 (i ∈ {1, …, ℓ}) with

S(K, …, K, M_{1}, …, M_{n−j−1}; ·) = ∑_{i=1}^{ℓ} a_{i} S_{n−1}(λ^{(i)}K + λ_{1}^{(i)}M_{1} + … + λ_{n−j−1}^{(i)}M_{n−j−1}; ·)

for all K, M_{1}, …, M_{n−j−1} ∈ K, compare the proofs of Hilfssatz 15.1 in [8] or of Lemma 5.1.3 in [9], for example. Now let K ∈ K and let M_{1}, …, M_{n−j−1} ∈ S. Define the signed measure

Θ_{j}(K; M_{1}, …, M_{n−j−1}; ·)
:= ∑_{i=1}^{ℓ} a_{i} t_{λ_{1}^{(i)}M_{1}+...+λ_{n−j−1}^{(i)}M_{n−j−1}}^{λ^{(i)}}(Θ_{n−1}(λ^{(i)}K + λ_{1}^{(i)}M_{1} + … + λ_{n−j−1}^{(i)}M_{n−j−1}; ·)).

It is concentrated on NorK and weakly continuous in K ∈ K. If K ∈ S, then

Θ_{j}(K; M_{1}, …, M_{n−j−1}; ·) = s_{K}(S(K, …, K, M_{1}, …, M_{n−j−1}; ·))

because of Θ_{n−1}(M; ·) = s_{M}(S_{n−1}(M; ·)) and t_{M}^{λ} ∘ s_{λK+M} = s_{K} for all M ∈ S, λ > 0. This shows that Θ_{j}(K; M_{1}, …, M_{n−j−1}; ·) is a positive measure, which is symmetric in M_{1}, …, M_{n−j−1}, and that the asserted relation (5) holds if K ∈ S. Now the general result follows by weak continuity. □

We are now in a position to formulate our extension of Theorem 3. Let f : S^{n−1} → R be a non-negative continuous function. Let δ > 0. Define a measure α on B(R^{n}) by means of

α(A) := ∫_{0}^{∞}t^{δ−1} ∫_{S^{n−1}} 1_{A}(tu)f(u) dσ(u)dt,

where σ is the spherical Lebesgue measure. Let ν be the invariant probability measure on the rotation group SO_{n}. Let μ_{α} be the image of the product measure α ⊗ ν on R^{n} × SO_{n} under the map R^{n} × SO_{n} → G_{n}, (x, ϑ) ↦ g_{x,ϑ}, where the motion g_{x,ϑ} is defined by g_{x,ϑ}(y) := x + ϑy. There is a unique convex body Z_{α} ∈ K with support function ∫_{B^{n}} |⟨x, ·⟩| dα(x), see [9], Theorem 1.7.1. A straightforward computation shows that this support function is of class C^{1}, so Z_{α} is strictly convex (cf. [9], Corollary 1.7.3). Let j ∈ {0, …, n − 1}, and let M_{1}, …, M_{n−j−1} ∈ S be strictly convex bodies. As before, we extend the domain of the measure Θ_{j}(K; M_{1}, …, M_{n−j−1}; ·), K ∈ K, by replacing this measure by its completion. Now the following can be stated.

Theorem 6. Under the above assumptions, we have

lim_{r→∞} \frac{1}{r^{δ+1}} ∫_{\{g∈G_{n}:gK′⊂rB^{n}\}} Θ_{j}(K ∨ gK′; M_{1}, …, M_{n−j−1}; (η ∩ NorK) ∨ g(η′ ∩ NorK′)) dμ_{α}(g)

= \frac{1}{2nκ_{n}} ∑_{k=0}^{j−1} Θ_{k}(K; M_{1}, …, M_{n−j−1}, B^{n}, …, B^{n}, Z_{α}; η)Θ_{j−k−1}(K′; η′)

uniformly for all η, η′ ∈ B(Σ) and all K, K′ ∈ K contained in a fixed ball.

We omit the proof of Theorem 6, since it follows exactly the same lines as the proof of Theorem 3 but requires a clumsier notation.

We remark that it is the continuity of f that insures the existence of the limit for all Borel sets η, η′, see the proof of Theorem 4 in [6]. Of course, Z_{α} is the zonoid whose generating measure has the density \frac{1}{δ+1}f with respect to σ.

Finally, we want to state a generalization of Theorem 1 without proof. This is a result in the spirit of Hadwiger’s general kinematic formula ([7], Section 6.3.5) and Schneider’s [10] local version of it (see [4], 105–108, for a simpler proof of the latter result).

Denote by E_{j}^{n} the set of all j-dimensional affine subspaces of R^{n}, endowed with its usual topology, and let μ_{j} be the Haar measure on E_{j}^{n}, normalized by μ_{j}({E ∈ E_{j}^{n} : E ∩B^{n} ≠ ∅}) = κ_{n−j}. For E ∈ E_{j}^{n} and η ⊂ Σ, we let

η ∧ E := {(x, u) ∈ Σ : there are u_{1}, u_{2} ∈ S^{n−1} with (x, u_{1}) ∈ η,
(x, u_{2}) ∈ E × E^{⊥}, and u ∈ pos{u_{1}, u_{2}}} ,

where E^{⊥} is the (n − j)-dimensional linear subspace orthogonal to E. Now the following can be stated.

Theorem 7. Let Ψ : K × B(Σ) → R be a map with the following properties.

1. Ψ(K; ·) is a measure, concentrated on NorK for all K ∈ K.

2. For K, K′ ∈ K, η ∈ B(Σ) with η ∩ NorK = η′ ∩ NorK′, we have Ψ(K; η) = Ψ(K′; η′).

3. The map K ↦ Ψ(K; ·) is weakly continuous.

Denote the completion of the measure Ψ(K; ·) by \overline{Ψ}(K; ·). For k ∈ \{1, …, n − 1\}, K ∈ K, η ∈ B(Σ), the integral

Ψ_{k}(K; η) := ∫_{\{E∈E_{n−k}^{n}:K∩E≠∅\}} \overline{Ψ}(K ∩ E; (η ∩ NorK) ∧ E) dμ_{n−k}(E)

exists, and we have

∫_{\{g∈G_{n}:K∩gK′≠∅\}} \overline{Ψ}(K ∩ gK′; (η ∩ NorK) ∧ g(η′ ∩ NorK′)) dμ(g)

= ∑_{k=1}^{n−1} \frac{\binom{n}{k}}{nκ_{k}} Ψ_{k}(K; η)Θ_{n−k}(K′; η′)

for all K, K′ ∈ K and all η, η′ ∈ B(Σ).

The proof of Theorem 7 combines techniques developed in [4], [5], and in the present paper.

Remarks. 1. Examples for the measure Ψ(K; ·) are the mixed support measures Θ_{j}(K; M_{1}, …, M_{n−j−1}; ·). In the case Ψ(K; ·) := Θ_{j}(K; ·), the Crofton formula proved in [5], Theorem 3.2, shows that Theorem 7 contains Theorem 1 as a special case.

2. It is easy to see that a map K ↦ Ψ(K; ·) satisfying conditions 1 and 2 of Theorem 7 is a valuation. Now one can apply Groemer’s extension theorem for continuous valuations (see, e.g., [12], Satz 2.4.2) to extend Theorem 7 to finite unions of convex bodies. See [5] for a definition of the set NorK if K is a finite union of convex bodies and for additional arguments required to establish this extension.

References

[1] Ash, R. B.: Measure, Integration, and Functional Analysis. Academic Press, New York 1972.

[2] Cohn, D. L.: Measure Theory. Birkhäuser, Boston 1980.

[3] Federer, H.: Curvature measures. Trans. Amer. Math. Soc. 93 (1959), 418–491.

[4] Glasauer, S.: Integralgeometrie konvexer Körper im sphärischen Raum. Thesis, University of Freiburg 1995. Summary: Diss. Summ. Math. 1 (1996), 219–226.

[5] Glasauer, S.: A generalization of intersection formulae of integral geometry. Geom. Dedicata 68 (1997), 101–121.

[6] Glasauer, S.: Translative and kinematic integral formulae concerning the convex hull operation. Math. Z. (to appear).

[7] Hadwiger, H.: Vorlesungen über Inhalt, Oberfläche und Isoperimetrie. Springer, Berlin 1957.

[8] Leichtweiß, K.: Konvexe Mengen. Springer, Berlin 1980.

[9] Schneider, R.: Convex Bodies: the Brunn-Minkowski Theory. Cambridge University Press, Cambridge 1993.

[10] Schneider, R.: An extension of the principal kinematic formula of integral geometry. Suppl. Rend. Circ. Mat. Palermo II 35 (1994), 275–290.

[11] Schneider, R.: Convex bodies in exceptional relative positions. J. London Math. Soc. (to appear).

[12] Schneider, R., Weil, W.: Integralgeometrie. Teubner, Stuttgart 1992.

[13] Schneider, R., Wieacker, J. A.: Integral geometry. In: Handbook of Convex Geometry (editors P.M. Gruber, J.M. Wills), North-Holland, Amsterdam 1993, 1349–1390.

Received October 31, 1997