Integral Geometry of Tensor Valuations

Integral Geometry of Tensor Valuations

Daniel Hug

∗ , Rolf Schneider

, Ralph Schuster

aUniversit¨at Duisburg-Essen, Campus Essen, Fachbereich Mathematik D-45117 Essen, Germany

bMathematisches Institut, Albert-Ludwigs-Universit¨at

Eckerstr. 1, D-79104 Freiburg, Germany

cD¨usseldorferstr. 2, D-80804 M¨unchen, Germany

Abstract

We prove a complete set of integral geometric formulas of Crofton type (involving in- tegrations over affine Grassmannians) for the Minkowski tensors of convex bodies. Min- kowski tensors are the natural tensor valued valuations generalizing the intrinsic volumes (or Minkowski functionals) of convex bodies. By Hadwiger’s general integral geometric theorem, the Crofton formulas yield also kinematic formulas for Minkowski tensors. The explicit calculations of integrals over affine Grassmannians require several integral geo- metric and combinatorial identities. The latter are derived with the help of Zeilberger’s algorithm.

MSC:52A20; 52A22; 53C65

Keywords: Convex body; intrinsic volume; tensor valuation; Minkowski tensor; integral geometry; Crofton formula; kinematic formula

1 Introduction

The kinematic formula, which goes back to Blaschke, Santal´o and Chern, is a major classical result of integral geometry. When restricted to convex bodies in Euclidean

? Supported in part by the European Network PHD, FP6 Marie Curie Actions, RTN, Con- tract MCRN-511953.

∗ Corresponding author.

Email addresses:daniel.hug@uni-due.de(Daniel Hug), rolf.schneider@math.uni-freiburg.de(Rolf Schneider), raschuster@munichre.com(Ralph Schuster).

space Rⁿ, it involves the intrinsic volumes (Minkowski functionals) V₀, . . . , V_n. These are determined as the coefficients in the Steiner formula

V_n(K+λBⁿ) =

n

X

i=0

κ_iVn−i(K)λⁱ, λ>0.

HereK is inKⁿ, the space of convex bodies inRⁿ,Bⁿdenotes the unit ball inRⁿ, of volumeκ_n, andV_nis the volume. In particular,Vn−1 is half the surface area,V₁ is proportional to the mean width, andV₀ is the Euler characteristic.

LetG(n)denote the motion group ofRⁿ, and letµbe the Haar measure onG(n), normalized as in [20, p. 227]. Then the kinematic formula for convex bodiesK, L∈ Kⁿsays that, forj ∈ {0, . . . , n},

Z

G(n)

V_j(K∩gL)µ(dg) =

n

X

k=j

α_njkV_k(K)V_n+j−k(L), (1.1) where

α_njk =

_k

j

κ_kκn+j−k

_n

k−j

κ_jκ_n .

Related to the kinematic formula is the Crofton formula, which involves an inte- gration over E_kⁿ, the affine Grassmannian of k-flats in Rⁿ. The motion invariant Haar measureµⁿ_k onE_kⁿwill be normalized so thatκn−kis the measure of the set of k-flats hittingBⁿ. Then, forK ∈ Kⁿ,k∈ {0, . . . , n}andj ∈ {0, . . . , k},

Z

E_kⁿ

V_j(K∩E)µⁿ_k(dE) =α_njkV_n+j−k(K). (1.2)

A basic result on intrinsic volumes is Hadwiger’s characterization theorem. It says that the vector space of continuous and motion invariant real valuations on Kⁿ is spanned byV0, . . . , Vn. This theorem can be used to prove (1.1) and (1.2). A deeper connection is exhibited by Hadwiger’s general integral geometric theorem, stating that for any continuous valuationϕ:Kⁿ→Rand forK, L∈ Kⁿone has

Z

G(n)

ϕ(K∩gL)µ(dg) =

n

X

q=0

ϕ_n−q(K)V_q(L) (1.3) with

ϕn−q(K) :=

Z

E_qⁿϕ(K∩E)µq(dE). (1.4) A modern presentation of the proof is found in [3, p. 132-4]. In particular, choosing ϕ = V_j in (1.3), we obtain (1.1) via (1.2). The remarkable fact is that for a gen- eral continuous valuationϕ, the kinematic integral on the left-hand side of (1.3) is known once the Crofton type integrals on the right-hand side of (1.4) have been determined.

During the last 30 years, the integral geometry of convex bodies has been general- ized considerably and extended in several directions; this includes local versions of classical results, more general translative formulas, and applications to stochastic geometry. A partial survey is given in [9].

One line of extension concerns the natural tensor valued generalizations of the in- trinsic volumes. The rank one case, vector valued valuations, was already inves- tigated in the 1970s by Hadwiger and Schneider, see [7], [19]. Characterizations and integral geometric formulas turned out to be very similar to the scalar case.

The systematic investigation of tensor valuations of higher rank, and in particular of Minkowski tensors, began only with a paper by P. McMullen [11], and here the situation turned out to be quite different.

To explain the Minkowski tensors, we denote byT^p the vector space of symmetric tensors of rank pover Rⁿ. We use the scalar product to identifyRⁿ with its dual space; thenT^pcan be viewed as the vector space of symmetricp-linear functionals onRⁿ. The symmetric tensor product of symmetric tensorsa, bis denoted by ab, andx^r is the r-fold symmetric tensor product ofx ∈ Rⁿ (and this is equal to the r-fold tensor product of x). We need the support measures (generalized curvature measures)Λ0(K,·), . . . ,Λn−1(K,·)of a convex bodyK ∈ Kⁿ, which are defined by a localized Steiner formula. Leth·,·ibe the scalar product andk · kthe norm in Rⁿ. The unit sphere ofRⁿis denoted bySⁿ⁻¹. Forx ∈ Rⁿ, let p(K, x)denote the metric projection ofxtoK, and putu(K, x) := (x−p(K, x))/kx−p(K, x)kfor x /∈K. Then, for any >0and Borel setη⊂Σ :=Rⁿ×Sⁿ⁻¹, then-dimensional Hausdorff measure (volume)Hⁿof the local parallel set

M(K, η) :={x∈(K+Bⁿ)\K : (p(K, x), u(K, x))∈η}

is a polynomial in,

Hⁿ(M(K, η)) =

n−1

X

k=0

^n−kκn−kΛ_k(K, η);

see [20], [21] for further information. Clearly, we have V_i(K) = Λ_i(K,Σ) for i= 0, . . . , n−1. In addition, we defineΛ_n(K,·)as the restriction ofHⁿtoK.

For K ∈ Kⁿ and integers r, s > 0, 0 ≤ j ≤ n−1, the Minkowski tensors are defined by

Φ_j,r,s(K) := 1 r!s!

ω_n−j ωn−j+s

Z

Σ

x^ru^sΛ_j(K,d(x, u)) and

Φ_n,r,0(K) := 1 r!

Z

Rⁿ

x^rΛ_n(K,dx),

whereωm := mκm. In view of later use, we extend the definition byΦj,r,s := 0if j /∈ {0, . . . , n}or ifrorsis not inN0or ifj =nands6= 0. Vector valuations are obtained forr = 1ands = 0; the caser = 0ands = 1leads to valuations which

are identically zero. All functionsΦ_j,r,s :Kⁿ →T^r+s are continuous (with respect to the Hausdorff metric on Kⁿ and the standard topology on T^r+s) and isometry covariant valuations (we refer to [11], [21], [10] for explicit definitions). These properties are still shared by the tensor functionsQ^mΦj,r,s,m∈N⁰, whereQ∈T² denotes the metric tensor, defined by Q(x, y) = hx, yifor x, y ∈ Rⁿ. We call the functionsQ^mΦ_j,r,s withr, s∈N0 and eitherj ∈ {0, . . . , n−1}or(j, s) = (n,0), thebasic tensor valuations(the terminology here and in [10] is different from the one in [21]).

The characterization theorems for rank zero by Hadwiger [6] and for rank one by Hadwiger and Schneider [7] were extended in remarkable work [1], [2] of Alesker, who obtained the following result.

Theorem 1.1 (Alesker) Letp∈ N0, and letϕ :Kⁿ → T^p be a continuous, isom- etry covariant valuation. Thenϕis a linear combination, with constant real coeffi- cients, of the basic tensor valuationsQ^mΦ_k,r,s, wherem, k, r, s∈N0 are such that 2m+r+s=p.

For generalp ∈N0, the basic tensor valuations of rankpare not linearly indepen- dent. McMullen [11] found that, forr>2andk ∈ {0, . . . , n+r−2},

2π^X

s

sΦk−r+s,r−s,s =Q^X

s

Φk−r+s,r−s,s−2. (1.5)

It was proved in [10] that these are essentially, that is, up to multiplication by pow- ers ofQand linear combinations, the only non-trivial linear relations between the basic tensor valuations. This led to the explicit determination of dimensions and bases for the vector spaces of continuous, isometry covariant tensor valuations of rankp. For example, for rank two the dimension is3n+ 1and a basis is given by

• QV_j,j = 0, . . . , n,

• Φ_j,2,0,j = 0, . . . , n,

• Φ_j,0,2,j = 1, . . . , n−1.

Since

Φ_k,0,2 = 1

4πQΦ_k,0,0− 1

2Φk−1,1,1,

for k = 1, . . . , n − 1, we can replace Φ_j,0,2, j = 1, . . . , n− 1, by Φ_j,1,1, j = 0, . . . , n−2, in the preceding basis. While the rank two case is still easy to treat, for tensor valuations of higher rank the situation is considerably more complicated.

The existence of non-trivial linear relations between the basic tensor valuations of rank p > 2 is one reason for the fact that, in contrast to the cases p = 0,1, the characterization theorem seems to be of no help in obtaining integral geometric formulas for Minkowski tensors of higher rank. Another reason lies in the difficulty of explicitly calculating Minkowski tensors for special convex bodies.

We mention that interest in the integral geometry of intrinsic volumes and their generalizations comes also from applied sciences. We refer to the work of K. Mecke [12], [13], [14] in statistical physics and point out that Minkowski tensors up to rank two are used in [5], [4], as tools in the morphometry of spatial patterns.

Basic integral geometry for Minkowski tensors requires to determine the kinematic integrals

Z

G(n)

Φ_j,r,s(K∩gL)µ(dg) (1.6)

and the Crofton integrals

Z

E_kⁿ

Φ_j,r,s(K∩E)µⁿ_k(dE). (1.7)

By Hadwiger’s general integral geometric theorem, which can be applied to each coordinate ofΦ_j,r,s with respect to some basis ofT^r+s, it is sufficient to determine the Crofton integrals (1.7). In some special cases, this has been done in [21]. For s = 0, for example, the required formulas follow immediately from the known corresponding formulas for curvature measures. The new cases settled in [21] are those where j = n − 1 or j = n − 2. Hence, all formulas of types (1.7) and (1.6) were obtained in dimensions n = 2 and n = 3. Further progress requires more sophisticated methods and calculations. The complete determination of all integrals (1.7) is the subject of this paper.

2 Results

For the statement of our main results, we distinguish two cases. This corresponds to the distinction between the casesk < nandk = nin the definition of the support measures. We start with the latter case which is easier to state and to prove.

Theorem 2.1 ForK ∈ Kⁿ,r, s∈N⁰ and06k 6n−1,

Z

E_kⁿ

Φ_k,r,s(K∩E)µⁿ_k(dE) =







˜

αn,k,sQ^2sΦn,r,0(K), ifsis even,

0, ifsis odd,

where

˜

αn,k,s = 1

(4π)^{s2 s}₂!

Γ^n−k+s₂ Γⁿ₂ Γ^n+s₂ Γ^n−k₂ .

Theorem 2.2 ForK ∈ Kⁿandk, j, r, s∈N⁰ with06j < k6n−1,

Z

E_kⁿ

Φ_j,r,s(K∩E)µⁿ_k(dE)

=

b^s₂c

X

z=0

χ⁽¹⁾_n,j,k,s,zQ^zΦn+j−k,r,s−2z(K) +

b₂^sc−1

X

z=0

χ⁽²⁾_n,j,k,s,zQ^z×

s−2z−1

X

l=0

(2πlΦn+j−k−s+2z+l,r+s−2z−l,l(K)−QΦn+j−k−s+2z+l,r+s−2z−l,l−2(K)),

where the constantsχ⁽¹⁾_n,j,k,s,z andχ⁽²⁾_n,j,k,s,zare given by(5.16)and(5.17).

The result can be given an alternative form by using the McMullen relations (1.5).

Theorem 2.3 ForK ∈ Kⁿandk, j, r, s∈N⁰ with06j < k6n−1,

Z

Eⁿ

k

Φ_j,r,s(K∩E)µⁿ_k(dE)

=

b^s₂c

X

z=0

χ⁽¹⁾_n,j,k,s,zQ^zΦn+j−k,r,s−2z(K) +

b^s₂c−1

X

z=0

χ⁽²⁾_n,j,k,s,zQ^z×

X

l>s−2z

(QΦn+j−k−s+2z+l,r+s−2z−l,l−2(K)−2πlΦn+j−k−s+2z+l,r+s−2z−l,l(K)),

with the same constants constantsχ⁽¹⁾_n,j,k,s,z andχ⁽²⁾_n,j,k,s,z as in Theorem 2.2.

In the preceding formulas, the Minkowski tensorΦ_j,r,s(K∩E)of the lower dimen- sional set K ∩E contained in E is computed inRⁿ. For convex bodies lying in an affine subspace E, there is an alternative version Φ^(E)_j,r,s of the Minkowski ten- sors, involving only support measures corresponding to that subspace; see the next section for a more detailed explanation. Formulas similar to those in the preceding theorems hold also ifΦj,r,s(K∩E)is replaced byΦ^(E)_j,r,s(K∩E).

Theorem 2.4 LetK ∈ Kⁿandk, r, s∈N0with06k 6n−1. Then

Z

E_kⁿΦ^(E)_k,r,s(K ∩E)µⁿ_k(dE) =







Φ_n,r,0(K), ifs= 0,

0, otherwise.

Theorem 2.5 LetK ∈ Kⁿandk, j, r, s∈N⁰ with06j < k6n−1. Then

Z

Eⁿ

k

Φ^(E)_j,r,s(K∩E)µⁿ_k(dE)

=

b^s₂c

X

z=0

¯

χ⁽¹⁾_n,j,k,s,zQ^zΦn+j−k,r,s−2z(K) +

b₂^sc−1

X

z=0

¯

χ⁽²⁾_n,j,k,s,zQ^z×

s−2z−1

X

l=0

(2πlΦn+j−k−s+2z+l,r+s−2z−l,l(K)−QΦn+j−k−s+2z+l,r+s−2z−l,l−2(K)),

where the constantsχ¯⁽¹⁾_n,j,k,s,z andχ¯⁽²⁾_n,j,k,s,zare given by(5.18)and(5.19).

Theorem 2.6 LetK ∈ Kⁿandk, j, r, s∈N0 with06j < k6n−1. Then

Z

Eⁿ

k

Φ^(E)_j,r,s(K∩E)µⁿ_k(dE)

=

b^s

2c

X

z=0

¯

χ⁽¹⁾_n,j,k,s,zQ^zΦn+j−k,r,s−2z(K) +

b^s

2c−1

X

z=0

¯

χ⁽²⁾_n,j,k,s,zQ^z×

X

l>s−2z

(QΦn+j−k−s+2z+l,r+s−2z−l,l−2(K)−2πlΦn+j−k−s+2z+l,r+s−2z−l,l(K)), with the same constants constantsχ¯⁽¹⁾_n,j,k,s,z andχ¯⁽²⁾_n,j,k,s,z as in Theorem 2.5.

3 Geometric identities for tensor valuations

In the proofs of our main results we shall use two identities for tensor valuations, which are due to McMullen [11]. The first of these relates the Minkowski tensors of convex bodies lying in some affine subspace to the Minkowski tensors computed in this subspace. Before stating these results, we introduce some notation.

ByLⁿ_k we denote the Grassmannian ofk-dimensional linear subspaces ofRⁿ. For L ∈ Lⁿ_k, k ∈ {0, . . . , n}, letp_L : Rⁿ → Ldenote the orthogonal projection, and defineπ_L :Sⁿ⁻¹\L^⊥ →L∩Sⁿ⁻¹by

π_L(u) := p_L(u) kp_L(u)k.

Fork =nthis is the identity map onSⁿ⁻¹, fork= 0we obtain the empty map.

The tensorQ(L) ∈ T² is defined byQ(L)(x, y) := hpL(x), pL(y)iforx, y ∈ Rⁿ. If (e₁, . . . , e_n) is an orthonormal basis of Rⁿ, we have Q = ^Pn_i=1e²_i. If now the basis is such that(e₁, . . . , e_k)is a basis ofL, thenQ(L) =^Pk_i=1e²_i.

ForE ∈ E_kⁿ, we denote by E⁰ the linear subspace which is a translate ofE. The linear subspaceE^⊥ := (E⁰)^⊥is the orthogonal complement ofE⁰.

LetE ∈ E_kⁿ, k ∈ {1, . . . , n}, and a convex bodyK ∈ Kⁿ withK ⊂ E be given.

For any j ∈ {0, . . . , k −1}, let Λ^(E)_j (K,·) denote thejth support measure ofK with respect toE. It is defined as the image measure of the restriction ofΛj(K,·) toRⁿ×(Sⁿ⁻¹\E^⊥)under the map

Rⁿ×(Sⁿ⁻¹\E^⊥)→Rⁿ×(E⁰∩Sⁿ⁻¹), (x, u)7→(x, π_E⁰(u)).

Thus the measureΛ^(E)_j (K,·)is defined on the Borel subsets ofRⁿ×(E⁰∩Sⁿ⁻¹), and it is concentrated onΣ^(E) :=E×(E⁰∩Sⁿ⁻¹)sinceK ⊂E. IfL∈ Lⁿ_k, which can be identified with R^k, then the restriction of Λ^(L)_j (K,·) toΣ^(L) is just thejth support measure ofK ⊂ L as defined inL. For a translation vectort ∈ Rⁿ, one hasΛ^(L+t)_j (K+t, η+t) := Λ^(L)(K, η), whereη+t :={(x+t, u) : (x, u)∈η}.

Next we define tensor valuations of convex bodies K contained in the affine sub- spaceE ∈ E_kⁿby

Φ^(E)_j,r,s(K) := ωk−j

r!s!ωk−j+s

Z

Σ^(E)

x^ru^sΛ^(E)_j (K,d(x, u)), (3.8) forj ∈ {0, . . . , k−1}, and

Φ^(E)_k,r,0(K) := 1 r!

Z

K

x^rH^k(dx); (3.9)

in all other cases,Φ^(E)_j,r,s is defined as the zero function. For polytopes, these tensor valuations can be described in a simple way. We denote byPⁿthe set of polytopes in Kⁿ, and for P ∈ Pⁿ and j ∈ {0, . . . , n}, F^j(P) is the set of j-dimensional faces ofP. The normal cone of the polytopeP at its faceF is denoted byN(P, F), and ifP is contained in the affine subspaceE, thenN_E(P, F) := N(P, F)∩E⁰. The proof of the following lemma is an immediate consequence of corresponding properties of the support measures (see [20, Section 4.2]).

Lemma 3.1 LetP ∈ Pⁿ,06j < nandr, s∈N0. Then Φ_j,r,s(P) = ^X

F∈F^j(P)

1 r!s!ωn−j+s

Z

F

x^rH^j(dx)

Z

N(P,F)∩Sⁿ⁻¹

u^sH^n−j−1(du).

ForE ∈ E_kⁿ,k ∈ {1, . . . , n}, a polytopeP ⊂E, and for06j < k, Φ^(E)_j,r,s(P) = ^X

F∈F^j(P)

1 r!s!ωk−j+s

Z

F

x^rH^j(dx)

Z

NE(P,F)∩Sⁿ⁻¹

u^sH^k−j−1(du).

The relation between these two types of tensors is not so simple as one might ex- pect.

Theorem 3.2 (McMullen) Let E ∈ E_kⁿ, k ∈ {0, . . . , n−1}, and K ∈ Kⁿ with K ⊂E. Letj ∈N0with06j 6k. Then, for allr, s∈N0,

Φ_j,r,s(K) = ^X

m>0

Q(E^⊥)^m

(4π)^mm!Φ^(E)_j,r,s−2m(K).

In the case of a linear subspace, this is Theorem 5.1 of McMullen [11]. From this case, the result for an affine subspace is obtained by developing(x+t)^r, for a fixed translation vectort, in the integrands on both sides.

To formulate another result by McMullen [11, p. 269], we need a bit more notation.

Let P ∈ Pⁿ be a polytope and F ∈ F^k(P) a k-face ofP, k ∈ {0, . . . , n}. For integersr, s∈N0, define

Υ_r(F) := 1 r!

Z

F

x^rH^k(dx) and

Θ_s(P, F) := 1 s!

Z

N(P,F)

x^se^−πkxk2H^n−k(dx).

Furthermore, putΥr(F) := 0wheneverr <0, andΘs(P, F) := 0whenevers <0 or ifk=nands 6= 0. Ifk < nands>0, then

Θ_s(P, F) = 1 s!

Z ∞ 0

λ^s+n−k−1e^−πλ2dλ

Z

N(P,F)∩Sⁿ⁻¹

u^sH^n−k−1(du)

= Γ(^s+n−k₂ ) s!2π^s+n−k2

Z

N(P,F)∩Sⁿ⁻¹

u^sH^n−k−1(du)

= 1

s!ω_s+n−k

Z

N(P,F)∩Sⁿ⁻¹

u^sH^n−k−1(du).

This yields a nice generalization of a well-known representation of intrinsic vol- umes, namely

Φ_k,r,s(P) = ^X

F∈F^k(P)

Υ_r(F)Θ_s(P, F)

forr, s∈N0andk∈ {0, . . . , n}; the casek =ncan be checked separately.

Lemma 3.3 (McMullen) Let P ∈ Pⁿ be a polytope. Then, for r, s ∈ N0 and k ∈ {0, . . . , n},

2πsΦ_k,r,s(P)

= ^X

F∈F^k(P)

Q(N(P, F))Υr(F)Θs−2(P, F) + ^X

G∈F^k+1(P)

Q(G)Υr−1(G)Θs−1(P, G), whereQ(N(P, F)) :=Q(linN(P, F))andQ(G) := Q((affG)⁰).

4 Integration over Grassmannians

In this preparatory section, we obtain some auxiliary results concerning integrations over Grassmannians, in particular transformation formulas of integral geometric type and the values of some special integrals.

We need some notation. The bracket or generalized sine function of two linear subspacesL, L⁰ ofRⁿ is defined as follows. One chooses an orthonormal basis of L∩L⁰ and extends it to an orthonormal basis ofL and to one ofL⁰. Then[L, L⁰] is the volume of the parallelepiped spanned by the obtained vectors. This depends only on the subspacesLandL⁰.

Forq, k ∈ {0, . . . , n}and a fixed subspaceF ∈ Lⁿ_k let

L^F_q :=







{L∈ Lⁿ_q :L⊂F}, ifq 6k, {L∈ Lⁿ_q :L⊃F}, ifq > k.

There exists a unique (Borel) probability measure ν_q^F on Lⁿ_q with the following properties: it is concentrated onL^F_q, invariant under SO(F)and SO(F^⊥), where SO(F)denotes the group of all rotations ofRⁿmappingF into itself and leaving F^⊥ pointwise fixed, andν_q^ϑF(ϑA) = ν_q^F(A)for every rotationϑ ofRⁿ and every Borel setA⊂ Lⁿ_q (see [22, Section 6.1]).

The transformation formula of the following lemma will later be used to sim- plify some integrations. It adapts an integration over all k-subspaces to a fixed r-subspace F with r + k > n, in the following way. One first integrates over all k-subspaces containing a fixed (r+k −n)-subspace of F, and then over all (r+k −n)-subspaces of F. The formula was proved, in a different but equiva- lent formulation, by Petkantschin [15, formula (48)]; it appears also in [18, formula (14.40)]. The caser =n−1is Lemma 5.6 in [8], with a different proof. The gen- eral case can also be obtained by a straightforward extension of the argument used there.

Lemma 4.1 Letk, r ∈ {0, . . . , n}withr+k > n, let h : Lⁿ_k → Rbe integrable andF ∈ Lⁿ_r. Then

Z

Lⁿ_k h(L)ν_kⁿ(dL) = c

Z

L^F_r+k−n

Z

L^U_k h(L)[F, L]^r+k−nν_k^U(dL)ν_r+k−n^F (dU), where

c:=





n−r

Y

j=1

Γ(^j₂) Γ(^{2n−r−k−j+1}₂ )









r+k−n

Y

j=1

Γ(^j₂) Γ(^r−j+1₂ )









k

Y

j=1

Γ(^j₂) Γ(^n−j+1₂ )





−1

.

We denote bylin{U, v}the linear hull ofU ⊂Rⁿandv ∈Rⁿ.

Corollary 4.2 Let u ∈ Sⁿ⁻¹, 1 6 k 6 n−1, and let h : Lⁿ_k → Tbe a tensor valued integrable function. Then

Z

Lⁿ_k

h(L)ν_kⁿ(dL) = ω_k 2ωn

Z

L^u_k−1^⊥

Z 1

−1

Z

U^⊥∩u^⊥∩Sⁿ⁻¹

|t|^k−1(1−t²)^n−k−22

×h(lin{U, tu+√

1−t²w})H^n−k−1(dw) dt ν_k−1^u⊥ (dU).

Proof.This follows from Lemma 4.1 by an application of the coarea formula. 2 We will have to calculate integrals of the form

Z

Lⁿ

k

[F, L]^aQ(L)^mν_kⁿ(dL)

whereF ∈ Lⁿ_r is fixed, m ∈ N⁰ anda > 0. As a first step, we consider the two special casesa = 0andm= 0.

Lemma 4.3 Form∈N0 andk ∈ {1, . . . , n},

Z

Lⁿ_k Q(L)^mν_kⁿ(dL) =

m−1

Y

j=0

k+ 2j

n+ 2jQ^m = Γ(^k₂ +m)Γ(ⁿ₂) Γ(ⁿ₂ +m)Γ(^k₂)Q^m.

Proof.The assertion is true fork = n. Suppose thatk 6 n−1. Since the tensor on the left-hand side is rotation invariant, it is a multiple ofQ^m. To determine the factor, fix any u ∈ Sⁿ⁻¹ and apply u^2m to the integral. By Lemma 4.2 we thus obtain

Z

Lⁿ_k

hQ(L)^m, u^2miν_kⁿ(dL)

= ω_k 2ω_n

Z

L^u_k−1^⊥

Z 1

−1

Z

U^⊥∩u^⊥∩Sⁿ⁻¹

|t|^k−1(1−t²)^n−k−22

×(hQ(U), u²i+htu+√

1−t²w, ui²)^mH^n−k−1(dw) dt ν_k−1^u⊥ (dU)

= ωk

2ω_n

Z

L^u_k−1^⊥

Z 1

−1

Z

U^⊥∩u^⊥∩Sⁿ⁻¹

|t|^k+2m−1(1−t²)^n−k−22 H^n−k−1(dw) dt ν_k−1^u⊥ (dU)

= ω_k 2ωn

Γ(^k₂ +m)Γ(^n−k₂ )

Γ(ⁿ₂ +m) ωn−k = π^k2Γ(ⁿ₂)Γ(^k₂ +m)2π^n−k2 2πⁿ²Γ(^k₂)Γ(ⁿ₂ +m)

= Γ(^k₂ +m)Γ(ⁿ₂) Γ(ⁿ₂ +m)Γ(^k₂),

which yields the assertion. 2

Lemma 4.4 Leta>0,r, k ∈ {1, . . . , n}withk+r >nandF ∈ Lⁿ_r. Then

A(n, k, r, a) :=

Z

Lⁿ_k[F, L]^aν_kⁿ(dL) =

n−r−1

Y

i=0

Γ(ⁿ⁻ⁱ₂ )Γ(^k−i+a₂ ) Γ(^n−i+a₂ )Γ(^k−i₂ ), where forr=nthe right-hand side is defined as1.

Proof.Forr=nork =nthe assertion holds. Suppose thatr, k ∈ {1, . . . , n−1}

withk+r>n. We fix someu₁ ∈Sⁿ⁻¹∩F^⊥. Then Lemma 4.2 yields

Z

Lⁿ_k

[F, L]^aν_kⁿ(dL)

= ω_k 2ω_n

Z

L^u

⊥ 1 k−1

Z 1

−1

Z

U^⊥∩u^⊥₁∩Sⁿ⁻¹

|t|^k−1(1−t²)^n−k−22 [F,lin{U, tu₁+√

1−t²w}]^a

× H^n−k−1(dw) dt ν_k−1^u⊥1 (dU). (4.10)

Let L ∈ Lⁿ_k and F be in general relative position and u₁ ∈/ L, L^⊥. Denoting by [F, L∩u^⊥₁]^(u⊥1)the bracket ofF andL∩u^⊥₁ calculated with respect tou^⊥₁, we have [F, L] = [F, L∩u^⊥₁]^(u⊥1)kp_L(u₁)k. (4.11) We now apply (4.11) with the linear subspaceL := lin{U, tu₁ +√

1−t²w} and t ∈(−1,1)(the assumption of general relative position is almost surely satisfied), and thus get

[F,lin{U, tu₁+√

1−t²w}] = [F, U]^(u⊥1)|t|.

Substituting the result into (4.10), we obtain

Z

Lⁿ_k[F, L]^aν_kⁿ(dL)

= ω_k 2ω_n

Z

L^u

⊥ 1 k−1

Z 1

−1

Z

U^⊥∩u^⊥∩Sⁿ⁻¹

|t|^k−1+a(1−t²)^n−k−22 [F, U]^(u⊥1)a

× H^n−k−1(dw) dt ν^u

⊥ 1

k−1(dU)

= ωk

2ω_n

Γ(^k+a₂ )Γ(^n−k₂ ) Γ(^n+a₂ ) ωn−k

Z

L^u

⊥ 1 k−1

[F, U]^(u⊥1)aν^u

⊥ 1

k−1(dU)

= Γ(ⁿ₂)Γ(^k+a₂ ) Γ(^k₂)Γ(^n+a₂ )

Z

L^u

⊥ 1 k−1

[F, U]^(u⊥1)aν^u

⊥ 1

k−1(dU).

This is a recursive formula. In the next step, we chooseu₂ ∈Sⁿ⁻¹ ∩F^⊥∩u^⊥₁ and

obtain

Z

Lⁿ_k[F, L]^aν_kⁿ(dL)

= Γ(ⁿ₂)Γ(^k+a₂ )Γ(ⁿ⁻¹₂ )Γ(^k−1+a₂ ) Γ(^k₂)Γ(^n+a₂ )Γ(^k−1₂ )Γ(^n−1+a₂ )

Z

L^u

⊥ 1∩u⊥

2 k−2

[F, U]^{(u⊥1∩u⊥2)a}ν_k−2^{u⊥1∩u⊥2} (dU).

The procedure terminates aftern−rsteps with the intgral

Z

L^u

⊥ 1∩...∩u⊥

n−r r+k−n

[F, U]^{(u⊥1∩···∩u⊥n−r)a}ν^u

⊥

1∩···∩u^⊥_n−r

r+k−n (dU) = 1, sincedim(u^⊥₁ ∩. . .∩u^⊥_n−r) =n−(n−r) = r= dimF. 2

The following proposition is a common generalization of the preceding two lem- mas. However, the proof of this extension is essentially based on the previous spe- cial cases.

Proposition 4.5 Leta >0,m∈N0,k, r∈ {1, . . . , n}withk+r>nandF ∈ Lⁿ_r. Then

Z

Lⁿ_k[F, L]^aQ(L)^mν_kⁿ(dL) = A(n, k, r, a)Γ(^n+a₂ ) Γ(^k+a₂ )Γ(^n+a₂ +m)

m

X

i=0

m i

!

Γ^k+a₂ +m−i

× Γ(^n−k₂ +i)Γ(^a₂ + 1)Γ(^r₂)

Γ(^n−k₂ )Γ(^a₂ + 1−i)Γ(₂^r +i)(−1)ⁱQ^m−iQ(F)ⁱ, whereΓ(−p)⁻¹forp∈N0 has to be read as0.

Proof. The case a = 0 is covered by Lemma 4.3. Hence we consider the case a > 0. We prove the assertion by induction with respect to the dimension of F^⊥. Forr=nwe have[F, L] = 1, and the assertion of the lemma follows from Lemma 4.3 and Lemma 6.3. This settles the case whereF has codimension0.

Let us assume that the assertion of the lemma is proved for dim(F^⊥) =n−r−1, r∈ {1, . . . , n−1}. We now establish the lemma for dim(F^⊥) = n−r. FixF ∈ Lⁿ_r and chooseu ∈ Sⁿ⁻¹ ∩F^⊥. Using Lemma 4.2 we get, as in the proof of Lemma 4.4,

Z

Lⁿ

k

[F, L]^aQ(L)^mν_kⁿ(dL)

= ωk

2ω_n

Z

L^u_k−1^⊥

Z 1

−1

Z

U^⊥∩u^⊥∩Sⁿ⁻¹

|t|^k−1(1−t²)^n−k−22 [F, U]^(u⊥)a|t|^a

×Q(U) + (tu+√

1−t²w)^2mH^n−k−1(dw) dt ν_k−1^u⊥ (dU).

Now we expand them-th power. Since the integrals of odd powers ofwvanish, we get

Z

Lⁿ_k[F, L]^aQ(L)^mν_kⁿ(dL)

= ω_k 2ωn

m

X

p=0

Z

L^u_k−1^⊥

Z 1

−1

Z

U^⊥∩u^⊥∩Sⁿ⁻¹

|t|^k+a−1(1−t²)^n−k−22 [F, U]^(u⊥)a

× m p

!

Q(U)^m−p(tu+√

1−t²w)^2pH^n−k−1(dw) dt ν_k−1^u⊥ (dU)

= ω_k 2ω_n

m

X

p=0 p

X

q=0

Z

L^u_k−1^⊥

Z 1

−1

Z

U^⊥∩u^⊥∩Sⁿ⁻¹

m p

! 2p 2q

!

|t|^k+a−1(1−t²)^n−k−22

×[F, U]^(u⊥)aQ(U)^m−pt^2p−2qu^2p−2q(1−t²)^qw^2qH^n−k−1(dw) dt ν_k−1^u⊥ (dU)

= ω_k 2ω_n

m

X

p=0 p

X

q=0

Z

L^u_k−1^⊥

Z 1

−1

Z

U^⊥∩u^⊥∩Sⁿ⁻¹

m p

! 2p 2q

!

|t|k+a−1+2p−2q

(1−t²)^{n−k2 −1+q}

×u^2p−2qw^2q[F, U]^(u⊥)aQ(U)^m−pH^n−k−1(dw) dt ν_k−1^u⊥ (dU).

The integration with respect towcan be carried out, since

Z

Sⁿ⁻¹

w^2qHⁿ⁻¹(dw) = 2ω_2q+n ω_2q+1Q^q. Integrating also with respect tot, we arrive at

Z

Lⁿ

k

[F, L]^aQ(L)^mν_kⁿ(dL)

= 1

√π Γ(ⁿ₂) Γ(^k₂)

m

X

p=0 p

X

q=0

Γ(^k+a₂ +p−q)

Γ(^n+a₂ +p) Γq+ ¹₂ m p

! 2p 2q

!

×

Z

L^u_k−1^⊥

[F, U]^(u⊥)aQ(U)^m−p(Q(u^⊥)−Q(U))^qν_k−1^u⊥ (dU)u^2p−2q.

Next we expand (Q(u^⊥)−Q(U))^q and change the order of the summation with respect toqto get

Z

Lⁿ_k[F, L]^aQ(L)^mν_kⁿ(dL)

= 1

√π Γ(ⁿ₂) Γ(^k₂)

m

X

p=0 p

X

q=0 p−q

X

j=0

Γ(^k+a₂ +q)

Γ(^n+a₂ +p)Γp−q+¹₂ m p

! 2p 2q

! p−q j

!

×(−1)^p−q−j

Z

L^u_k−1^⊥

[F, U]^(u⊥)aQ(U)^m−q−jν_k−1^u⊥ (dU)Q(u^⊥)^ju^2q.

SinceF has codimensionn−1−rwith respect tou^⊥, we can apply the induction hypothesis. This yields

Z

Lⁿ

k

[F, L]^aQ(L)^mν_kⁿ(dL)

= 1

√π Γ(ⁿ₂) Γ(^k₂)

m

X

p=0 p

X

q=0 p−q

X

j=0

Γ(^k+a₂ +q)

Γ(^n+a₂ +p)Γp−q+ ¹₂ m p

! 2p 2q

! p−q j

!

×(−1)^p−q−jQ(u^⊥)^ju^2q A(n−1, k−1, r, a)Γ(^n−1+a₂ ) Γ(^k−1+a₂ )Γ(^n−1+a₂ +m−q−j)

m−q−j

X

i=0

m−q−j i

!

× Γ(^k−1+a₂ +m−q−j−i)Γ(^n−k₂ +i)Γ(^a₂ + 1)Γ(₂^r) Γ(^n−k₂ )Γ(^a₂ + 1−i)Γ(^r₂ +i)

×(−1)ⁱQ(u^⊥)^{m−q−j−i}Q(F)ⁱ.

We change the order of summation and do some rearrangements to get

Z

Lⁿ

k

[F, L]^aQ(L)^mν_kⁿ(dL)

= A(n−1, k−1, r, a)

√π

Γ(ⁿ₂)Γ(^n−1+a₂ ) Γ(^k₂)Γ(^k−1+a₂ )

m

X

i=0

Γ(^n−k₂ +i)Γ(^a₂ + 1)Γ(^r₂) Γ(^n−k₂ )Γ(^a₂ + 1−i)Γ(^r₂ +i)

×(−1)ⁱQ(F)ⁱ





m−i

X

q=0

u^2qQ(u^⊥)^m−q−iΓ^k+a₂ +q





m−q−i

X

j=0

m−q−j i

!

× Γ(^k−1+a₂ +m−q−j−i) Γ(^n−1+a₂ +m−q−j)





m

X

p=j+q

Γ(p−q+ ¹₂) Γ(^n+a₂ +p)

m p

! 2p 2q

!

× p−q j

!

(−1)^p−q−j

!!!

.

Then we first apply Lemma 6.1 to simplify the summation overp, and subsequently use Lemma 6.2 to simplify the summation overj. Thus we arrive at

Z

Lⁿ_k[F, L]^aQ(L)^mν_kⁿ(dL)

= A(n−1, k−1, r, a)Γ(ⁿ₂)Γ(^a₂ + 1)Γ(₂^r) Γ(^k₂)Γ(^n+a₂ +m)Γ(^n−k₂ )

m

X

i=0

Γ(^n−k₂ +i) Γ(^a₂ + 1−i)Γ(₂^r +i)

×(−1)ⁱQ(F)ⁱ m i

!

Γ^k+a₂ +m−i





m−i

X

q=0

u^2qQ(u^⊥)^m−q−i m−i q

!

.

The expression in brackets is just(Q(u^⊥) +u²)^m−i =Q^m−i. Since also A(n, k, r, a) = Γ(ⁿ₂)Γ(^k+a₂ )

Γ(^k₂)Γ(^n+a₂ )A(n−1, k−1, r, a), the induction is finished. 2

Here we are mainly interested in the special casea= 2.

Corollary 4.6 Letk, r∈ {1, . . . , n}withk+r>n,F ∈ Lⁿ_r andm ∈N0. Then

Z

Lⁿ_k

[F, L]²Q(L)^mν_kⁿ(dL) = k!r!

(k−n+r)!n!

Γ(ⁿ₂ + 1)Γ(^k₂ +m) Γ(ⁿ₂ +m+ 1)Γ(^k₂ + 1)

×^k₂ +mQ^m+^m_r(k−n)Q^m−1Q(F).

Proof.The proof follows from Lemma 4.4 and Proposition 4.5. 2

The mean value formula of Corollary 4.6 has to be refined further. We introduce the following constants, which are needed in the subsequent proposition and its proof:

β_n,j,k := (k−1)!(n+j−k)!

√πj!(n−1)!

Γ(ⁿ₂)Γ(ⁿ⁺¹₂ ) Γ(^k₂)Γ(^k+1₂ ), γ_n,k,l,p,q⁽¹⁾ :=

q

X

y=0

(−1)^l+y q y

!Γ(^k−1₂ +l−p+y) Γ(ⁿ⁺¹₂ +l−p+y)

_k−1

2 +l−p+y,

γ_n,k,l,p,q⁽²⁾ :=

q

X

y=0

(−1)^l+y q y

!Γ(^k−1₂ +l−p+y)

Γ(ⁿ⁺¹₂ +l−p+y)(l−p+y) withγ_n,k,l,p,q⁽²⁾ = 0ifl−p+q= 0. Moreover, we define

ζn,j,k,s,z,m⁽¹⁾ :=

m

X

l=max{0,m−z}

l

X

p=0

b^s₂c−m+p

X

q=max{0,z−m+p}

(−1)^m−p+q−zγ_n,k,l,p,q⁽¹⁾ m l

! l p

!

× s−2m+ 2p 2q

! l−p+q z−m+l

!Γ(^s+j₂ −m+p−q+ 1)Γ(q+¹₂) Γ(^s+n−k+j₂ −m+p+ 1)