Additive kinematic formulas for flag area measures

We show the existence of additive kinematic formulas for general flag area measures, which generalizes a recent result by Wannerer. Building on previous work by the second named author, we introduce an algebraic framework to compute these formulas explicitly. This is carried out in detail in the case of the incomplete flag manifold consisting of all $(p+1)$-planes containing a unit vector.

1. Introduction 1.1. Global and local kinematic formulas. Let V be a Euclidean vector space of dimension n and unit sphere S n−1 . Let K(V ) denote the space of compact convex subsets in V . A valuation is a finitely additive map on K(V ), that is, a map µ : K(V ) → A, where A is any abelian semigroup, and such that µ(K ∪ L) + µ(K ∩ L) = µ(K) + µ(L), whenever K, L, K ∪ L ∈ K(V ). If A is a topological semigroup, then continuity of µ is understood with respect to the Hausdorff topology on K(V ).
For the sake of brevity, we will call a group from this list a transitive group. The computation of the dimension as well as an explicit geometric description of a basis of Val G in each of these cases is a challenging problem and many results have been obtained recently by various authors [2,5,6,7,10,14,15,16].
Federer's curvature measures C 0 , . . . , C n are valuations with values in the space of signed measures on V . We refer to Schneider [35] for a classification result of these curvature measures. Fu [21] and Bernig-Fu-Solanes [11] studied smooth curvature measures in a broader context. For a transitive group G, there are local kinematic formulas for translation and G-invariant smooth curvature measures. The case G = SO(n) was known by Federer, whereas the hermitian case G = U(n/2) was described completely in [10,11,12].
The classical surface area measures S 0 , . . . , S n−1 are valuations with values in the space of signed measures on the unit sphere. Additive kinematic formulas for surface area measures have been shown in [34]. Smooth area measures which are equivariant with respect to a transitive group G were introduced by Wannerer [40,41], who proved the existence of local additive kinematic formulas and established such formulas in the hermitian case. Since these results are very influential for the present work, let us state them explicitly. Theorem 1.1. Let G be a transitive group. Then the space Area G of smooth, G-invariant area measures is finite-dimensional. If Φ 1 , . . . , Φ N is a basis of Area G , then there are local additive kinematic formulas where K, L ∈ K(V ) and where κ and λ are Borel subsets of S n−1 .

The linear map
A : Area G → Area G ⊗ Area G , Φ i → c i k,l Φ k ⊗ Φ l is a cocommutative, coassociative coproduct. The transposed map A * : Area G, * ⊗ Area G, * → Area G, * thus provides Area G, * with the structure of a commutative associative algebra.
In the case G = SO(n), this algebra is isomorphic to R[t]/ t n . The case G = U(n/2) is more involved. Using results from hermitian integral geometry [10] and from the theory of tensor valuations [13], Wannerer could write down this algebra in an explicit way. where I n is the ideal generated by f n/2+1 (s, t), f n/2+2 (s, t), p n/2 (s, t) − q n/2−1 (s, t)v, v 2 with log(1 + tx + sx 2 ) = ∞ k=0 f k (s, t)x k Although it is possible to write down the local additive kinematic formulas for G = U(n/2) using this theorem, the result is not as explicit as one would like. In [8], the second named author has shown that the algebra structure on Area G, * is induced from the algebra structure of a larger (infinite-dimensional) space Area * ,sm of smooth dual area measures and that the product in this algebra can be computed in some easy, algorithmic way. This has led to very explicit local additive kinematic formulas.

1.2.
Results of the present paper. Let V be an n-dimensional Euclidean vector space with unit sphere S n−1 . Let G be a subgroup of O(n) acting transitively on the sphere S n−1 and let G : is the stabilizer of the first basis vector. Then M := G/H is a G-homogeneous manifold.
A smooth flag area measure is a translation invariant valuation with values in the space of signed measures on M = G/H which is given by a certain smooth differential form on V × M . We denote by Area G/H the space of smooth flag area measures and by Area G G/H the subspace of smooth flag area measures which are equivariant with respect to the action, i.e., We refer to Section 2 for the complete definition. Our first two main theorems generalize [ where κ, λ are Borel subsets of M . Examples: (i) Let G be a subgroup of O(n) which acts transitively on S n−1 . Let H = G ∩ O(n − 1) be the stabilizer of the action. Then M = S n−1 and the flag area measures in this case are called area measures. The kinematic formulas in the case G = O(n) are classical [34]. For more general G, Wannerer has shown the existence of kinematic formulas [40, Theorem 2.1] and our proof follows his arguments. Explicit formulas in the hermitian case G = U(n) are contained in [8,39,40].
. Some elements in the space FlagArea (p),SO(n) := Area G G/H were constructed using a Steiner type formula in [28], see also [24]. A complete description of this space was given in [1]. (iii) If G ⊂ SO(n) and H = {1}, we also call the elements in Area G G/{1} rotation measures. They will be studied in Section 4.
The local additive kinematic formulas can be encoded by the map which is a cocommutative, coassociative coproduct. Alternatively, the dual space (Area G, * G/H , A * ) is a commutative, associative algebra. We give an explicit construction and classification of rotation measures. Consider a smooth compact convex body and x ∈ ∂K. Let ν(x) be the outer normal vector at x. Given g ∈ SO(n) with ge 1 = ν(x), the vectors ge 2 , . . . , ge n span T x ∂K. The shape operator is the self-adjoint linear map such that for every compact convex body with smooth boundary (1) Here V I = span{ge i , i ∈ I}, V J := span{ge j , j ∈ J} are oriented k-dimensional subspaces of T x ∂K and π J ⊥ : T x ∂K → V ⊥ J is the orthogonal projection. There are linear relations among these rotation measures: S I,J is antisymmetric in I and antisymmetric in J. Moreover, given In the important case of the incomplete flag manifold consisting of pairs (v, E) with dim E = p + 1, v ∈ E, we write down the algebra structure more explicitly.
where deg x = deg y = 1. If p = q = n−1 2 , then there is a graded isomorphism A basis of the space FlagArea if p = q = n−1 2 ), see [1]. The algebra structure given in the previous corollary translates into explicit kinematic formulas for these flag area measures.
Let ω n denote the volume of the (n − 1)-dimensional unit sphere and let c n,k,p,i := n − 1 k as in [1]. For given p, q, k, we write m k := min{p, q, k, n − k − 1}, m ′ k := min{p, k} and define We do not know whether this expression can be simplified any further.
Theorem 4 (Local additive kinematic formulas for flag area measures). Let 0 ≤ p, k ≤ n − 1, and 0 ≤ i ≤ m k . Then, Acknowledgments. We thank Thomas Wannerer for useful comments on a first draft of this paper.

Existence of kinematic formulas for smooth flag area measures
In this section we will introduce smooth area measures. Our definition will be justified by the existence of kinematic formulas for smooth area measures that will be shown in this section.
2.1. Fiber integration. We first collect some definitions and results from the theory of fiber bundles that will be used in the following. Definition 2.1. Let (E, π, B, F ) be a fiber bundle, with B and E oriented manifolds and F compact with dim F = r. The fiber integration or push- for every differential form ω ∈ Ω * (B) with compact support.
We then have the projection formula We note that for the above definition of fiber integration, we follow the sign convention in [3], as in [1,40]. For another sign convention see, e.g., [4]. (7)]). Let (E, π, B, F ) be a fiber bundle, with B orientable and E oriented with the local product orientation. If N ⊂ B is a compact and oriented submanifold with dim N = n and π −1 (N ) ⊂ E has the local product orientation, then, for every ω ∈ Ω n+r (E) with fiber-compact support,

2.2.
Smooth flag area measures. Let us introduce our main object of study. Let e 1 be the first standard vector in R n and let O(n − 1) be its stabilizer. Let M := G/H be a homogeneous space, where G is a closed subgroup of O(n) and where H is a closed subgroup of G ∩ O(n − 1). Let Π : V × M → SV, (x, gH) → (x, ge 1 ), which is a fiber bundle. We let r denote the dimension of the fiber and m := n − 1 + r.
where nc(K) denotes the normal cycle of K. The space of smooth flag area measures is denoted by Area G/H , and Area G G/H denotes the subspace of smooth flag area measures equivariant under the action of G given by and K ∈ K(V ), we obtain, by using (3), Hence globΦ is represented by the formΠ * ω .
The space Ω l (V × M ) tr of translation invariant forms admits a filtration as follows. Then

Kinematic formulas.
If G is transitive, then the space of smooth flag area measures is a quotient of the finite-dimensional space Ω m (V × M ) G and hence finite-dimensional itself. If G is not transitive, then Val G is not finite-dimensional. Since we have a surjective map glob : Area G G/H → Val G , dim Area G G/H = ∞ as well in this case. By using the definition of smooth flag area measures, we can restate the remaining part of Theorem 1 as follows.
Proof of Theorem 2.9. We follow the ideas of Fu [21] and Wannerer [40]. We first assume that the convex bodies K and L have smooth boundaries.
We define and the maps We define E ′ together with maps p ′ : E ′ → SV × SV, q ′ : E ′ → G × SV in an analogous way, using S n−1 instead of M . We then have a map The actions on SV × SV, E ′ , and G × SV are defined analogously. Then the maps in the above diagram areḠ ×Ḡ-equivariant.
The normal cycle K + gL is given by Applying the pull-back Π * , Lemma 2.4 yields We set F : Using the previous computation and Lemmas 2.2 and 2.3 (applied to The general case of not necessarily smooth convex bodies follows by approximation as in [40].

Dual flag area measures
In this section we introduce the notion of smooth dual flag area measure, which generalizes the notion of smooth dual area measure from [8]. Similarly to the case of dual area measures, we define a convolution product on the space of smooth dual area measures.
The space Ω m (V × M ) tr of translation invariant forms is endowed with the usual Fréchet topology of uniform convergence on compact subsets of all partial derivatives and there is a surjection Ω m (V × M ) tr → Area G/H . We endow the latter space with the quotient topology and denote by Area * G/H the dual space to Area G/H .
Analogously to the case of area measures, we define a convolution product on a subspace of Ω n (V × M ) tr .
First, we introduce an operator on the space of differential forms on V ×M which will play the analogous role, and is defined analogously, to * 1 from [9] and [8]. We denote this operator again by * 1 .
is closed under the operation The other two conditions can be proved as in the case p = 0 (see [8]).
Then (π 1 ) * τ ∈ Ω n (V ) tr is a multiple of the Lebesgue measure. We denote this multiple by τ .
Definition 3.4. A dual flag area measure L ∈ Area * G/H is called smooth if there exists τ ∈ J n,tr such that Proof. By Lemma 2.7 and Definition 2.1, we have Definition 3.6. Let L 1 , L 2 ∈ Area * ,sm G/H be represented by forms τ 1 , τ 2 ∈ J n,tr . Then we define L 1 * L 2 ∈ Area * ,sm G/H as the smooth dual area measure represented by τ 1 * τ 2 = * −1 1 ( * 1 τ 1 ∧ * 1 τ 2 ) ∈ J n,tr . Theorem 3.7. Let G ⊂ O(n) be a closed subgroup acting transitively on the unit sphere and H ⊂ G ∩ O(n − 1) be a closed subgroup. Then the following diagram commutes Here q G is the map transposed to the inclusion Area G G/H ֒→ Area G/H . We need some preparation before proving the theorem.
Proof. Let µ ∈ (Val ⊗Γ) G . By Alesker's irreducibility theorem [2] we may approximate µ by a sequence µ i ∈ Val sm ⊗Γ. Averaging with respect to Haar measure on G we find an approximating sequenceμ i ∈ (Val sm ⊗Γ) G . But the latter space is finite-dimensional, since it is a quotient of the space of translation-and G-invariant, Γ valued smooth differential forms, which is obviously finite-dimensional. It follows that µ belongs to (Val sm ⊗Γ) G . Proposition 3.9 (Kinematic formulas for tensor valuations). Let (Γ i , ρ i ) be finite-dimensional G-modules. If µ 1 , . . . , µ k is a basis of (Val ⊗Γ 1 ) G and φ 1 , . . . , φ l is a basis of (Val ⊗Γ 2 ) G , then for every τ ∈ (Val ⊗Γ 1 ⊗ Γ 2 ) G there are constants c τ ij such that We thus obtain a cocommutative coassociative coproduct Proof. This follows from the usual Hadwiger argument, compare [13, Section 3.2] for a similar situation.
We remark that there is another version of Poincaré duality which uses the Alesker product instead of the convolution. Following [40] we put a hat to distinguish between the two dualities.
In the particular case Γ = Sym l V , the moment map was already used in [40] and [8].
Lemma 3.16. The additive kinematic formulas are compatible in the following sense. Let Γ 1 , Γ 2 ⊂ C ∞ (G) be finite-dimensional submodules and let Γ 1 ·Γ 2 be the (finite-dimensional) module generated by all the natural projection, Then the following diagram commutes Proof of Theorem 3.7. We look first at rotation measures. Let Γ 1 , Γ 2 ⊂ C ∞ (G) be two finite-dimensional G-submodules. Dualizing the above diagram yields the commutative diagram . This means that if L 1 ∈ Area G * G/{1} is in the image of ( pd • M Γ 1 ) * and L 2 is in the image of ( pd • M Γ 2 ) * , then the formula to compute L 1 * L 2 is correct. Thus, by Lemma 3.14 the formula holds for all L 1 , L 2 ∈ Area G * G/{1} . In the general case, we have a commutative diagram Since glob * is injective, the statement follows from Lemma 3.5 and the fact thatΠ * commutes with * 1 .

Rotation measures
In this section we consider rotation measures, i.e., the case G := SO(n), H := {1} and prove Theorems 2 and 3.

Classification of rotation measures.
Proof of Theorem 2. In the first part of the proof, we follow [1, Section 3].
The Lie algebras of G, G will be denoted byḡ, g. The dimension of the fiber of Π : V ×G → SV is given by r := n−1 2 . As above we set m := n−1+ r. By definition, a smooth flag area measure Φ ∈ Area SO(n) SO(n)/{1} of degree k is represented by a translation invariant differential form η ∈ Ω k,m−k (V × G). Since Φ is G-invariant, we may assume by averaging over G that η is Ginvariant, i.e., η ∈ Ω m (Ḡ)Ḡ.
Let σ i , i = 1, . . . , n, ω ij , 1 ≤ i, j ≤ n denote the components of the Maurer-Cartan form ofḠ. Then σ i , 1 ≤ i ≤ n, ω i,j , 1 ≤ i < j ≤ n spanḡ * . We let X i , 1 ≤ i ≤ n, X ij , 1 ≤ i < j ≤ n denote the dual basis of g. We let V 0 be the span of X 1 ; V σ the span of X i , 2 ≤ i ≤ n; V ω be the span of the X 1j , 2 ≤ j ≤ n; and U the span of the X ij , 2 ≤ i < j ≤ n. Schematically, the Lie algebra looks as follows: Since Π * α = σ 1 , the quotient of the space ofḠ-invariant forms of bidegree (k, m − k) by multiplies of Π * α is the space If η belongs to the sum of terms with i < dim U * = r, then η ∈ F m,r and hence induces the trivial flag area measure. We thus obtain that Next, by Proposition 2.8, we have to factor out multiples of Π * dα, so that In particular, dim Area We now construct the rotation measures S I,J . Uniqueness follows from the fact that smooth convex bodies are dense in the space of all compact convex bodies. Let us prove existence. Let I = (i 1 , . . . , i k ), J = (j 1 , . . . , j k ) and set I c = {2, . . . , n} \ I, ordered in such a way that sgn(2, . . . , n) = sgn(i 1 , . . . , i k , i c 1 , . . . , i c n−k−1 ) and similar for J c . Define S I,J by the differential form where ρ is the volume form of the fiber of the map SO(n) → S n−1 , g → ge 1 . Let K be a smooth compact convex body with outer unit normal ν : ∂K → S n−1 . Fix x ∈ ∂K and g ∈ SO(n) with ge 1 = ν(x). Then ge 2 , . . . , ge n form a positive orthonormal basis of T x ∂K. The vectors w l := (ge l , S x (ge l )) ∈ T x ∂K × T x ∂K, l = 2, . . . , n span T (x,ν(x)) nc(K).
As above, we have The condition Π * α ∧ τ = 0 is satisfied for the part with ǫ = 1 in this sum, and the condition τ ∈ F n,1 is equivalent to i = 0. We are thus left with the space

Multiplication by the symplectic form dα gives a surjection
and Area SO(n) * SO(n)/{1},k is isomorphic to the kernel of this map. Let us rewrite this in more invariant terms. Let V C σ := V σ ⊗ C stand for the complexification. The group SL(n − 1, C) acts on the set of bases of V C σ , V C ω from the right in the natural way. If Y = (Y 2 , . . . , Y n ) is a basis of V C σ and g ∈ SL(n − 1, C), then (Y g) i := n j=2 Y j g ji . The corresponding right operation on V * ,C σ , V * ,C ω is given by As SL(n − 1, C)-representations, we have V C σ ∼ = (V C ω ) * . The symplectic form is the canonic element of Λ 2 (V C ω ⊕ (V C ω ) * ). The above map L can then be rewritten as an SL(n − 1, C)-equivariant surjection and with this identification, the map L : Let L k be the kernel of this map. It is well-known that L k is an irreducible representation of SL(n − 1, C) [22,Exercise 15.30]. Moreover, L k ·L l ⊂ L k+l .
(ii) We prove this by induction over n, the case n = 2 being trivial. Suppose that n > 2 and develop the determinant with respect to the last column: Now develop the determinant in the first summand with respect to the last row. We obtain some sum of terms containing the factor x l,n x nj , which equals − 1 2 x n,n x lj . We may thus replace the factor x l,n by − 1 2 x n,n , and the last row of (x ij ) 2≤i≤n,i =l 2≤j≤n−1 by x l2 , . . . , x l,n−1 , which is just the row which was deleted. Rearranging the rows (which gives us another sign (−1) n−l+1 ) we find that (iii) By definition, where I = (a 1 , . . . , a i ) runs over all ordered subsets of size i in {2, . . . , n}. Since x 2 aa = 0 for each a, we find that E i (x 2,2 , . . . , x n,n )E j (x 2,2 , . . . , x n,n ) = |I|=i,|J|=j There are n−1 k−l possibilities to choose the (k − l) double indices among 1, . . . , n − 1. Replacing x ij 1 x ij 2 by − 1 2 x ii x j 1 j 2 , we may assume that such a double index i appears in a factor x ii .
From the remaining (n − k + l − 1) other indices we choose 2l, which gives us n−k+l−1 2l possibilities. These 2l indices i 1 , . . . , i 2l will be put into pairs so that we form the product x i 1 i 2 · · · x i 2l−1 i 2l .
However, we can use the relations to rule out some combinations. Arrange the 2l numbers i 1 , . . . , i 2l in a circle. If a monomial contains a factor x i 1 i 2 x i 3 i 4 such that the lines between [i 1 , i 2 ] and [i 3 , i 4 ] intersect, we may use the relation and replace this factor by (l+1)!l! . Summarizing, we get that the dimension of the degree k-part of the algebra is bounded from above by It remains to see that this equals the expression given in the lemma. This can be seen by the following combinatorial argument.
Take a set of n numbered cards with both sides empty. Choose k among these cards and color the front side green. Independently of that, color the back side of (k+1) among the n cards red. The number of different colorings obtained in this way is n k n k+1 . Start again with a set of n numbered cards with both sides empty and fix a number 0 ≤ l ≤ k. Choose one of the cards and declare it to be 1-colored (the color will be fixed later). Among the remaining (n − 1) cards, choose (k − l) and color the front side green and the back side red. Among the remaining (n − k + l − 1) cards, choose 2l and declare them to be 1-colored. Among the (2l + 1) 1-colored cards, we color l in green and (l + 1) in red. In this way, we obtain all colorings with precisely (k − l) two-colored cards, such that k are green and (k + 1) are red. However, each of these colorings is counted (2l +1) times, since any of the (2l +1) 1-colored cards can be chosen as the first card to begin with. The total number of colorings is therefore n k It follows that In both cases, it follows that Ψ(  Proof. The statement follows directly by recalling that the inclusion glob * preserves the algebra structure of both spaces and is H-equivariant. Since the dimensions on both sides agree, this map is an algebra isomorphism. Hence, which is of course well-known. The additive kinematic formulas are given by see [13], or Schneider [36,Theorem 4.4.6]. It follows that Clearly t is mapped to some multiple cS * 1 . We will see later that c = ω n . Then t k is mapped to ω n k!S * k , as can be shown by induction over k.

5.
Algebraic structure of FlagArea (p),SO(n), * The aim in this section is to prove Proposition 1.3. Recall first that σ i , ω ij are the coordinates of the Maurer-Cartan form of SO(n). The volume form of the unit sphere is ω 21 ∧ . . . ∧ ω n1 . We have the structure equations Unwinding the definitions in Section 4, we have which sends x i y j to i!E i (x 2,2 , . . . , x p+1,p+1 )j!E j (x p+2,p+2 , . . . , x n,n ) is an algebra morphism. It is clear that the image of each monomial is H-invariant. The compatibility with the product follows from Lemma 4.1.
Obviously, x p+1 , y q+1 ∈ ker Ξ, hence there is an induced algebra morphism We claim that this map is injective. To do so, introduce a bigrading on (R[X]/I) H by declaring that It is easily checked that the ideal I is bigraded, so we indeed have a bigrading on the quotient. The image of x i y j , 0 ≤ i ≤ p, 0 ≤ j ≤ q is of bidegree (i, j).
To prove injectivity ofΞ, it is therefore enough to prove thatΞ(x i y j ) = 0 for 0 ≤ i ≤ p, 0 ≤ j ≤ q. ButΞ(x i y j ) corresponds to the form (10) i which is obviously non zero.
To conclude the proof thatΞ is an algebra isomorphism, it is enough to compare dimensions. The dimension of the k-homogeneous part of the left hand side is the number of monomials x i y j with i + j = k, 0 ≤ i ≤ p, 0 ≤ j ≤ q, which is easily computed as min{p, q, k, n−k−1}+1. The k-homogeneous part on the right hand side is isomorphic to FlagArea (p),SO(n) k , which is of the same dimension by [1,Theorem 4].
Let us now consider the case p = q. The above proof goes through word by word, except that dim FlagArea We next compute Ξ(u 2 ) by using x ai x aj = − 1 2 x aa x ij , which is a consequence of Lemma 4.1.
We denote (13) ω k,a := ω n vol(Flag 1,p+1 )η k,a ∧ρ ∈ Ω m (V ×Flag 1,p+1 ), max{0, k−q} ≤ a ≤ min{k, p}, and, if n is odd and 2p = n − 1, Here ρ denotes the volume form of the fiber of the map Π : V × Flag 1,p+1 → V × S n−1 . We remark that the factor in the definition ofω k,a does not appear explicitly in [1], but implicitly by the fact that the volume form on the fiber should be normalized to volume 1 (see [1,Corollary 4.6]).
The proof is finished by noting thatΦ * ex = u.

Explicit additive kinematic formulas
The aim in this section is to obtain explicit additive kinematic formulas for FlagArea (p),SO(n) . We denote by (ii) The additive kinematic formulas for SO(n) are given as follows. If We have It follows that c k,a j,b,l,c equals zero unless k = j + l and a = b + c and the result follows.
Let us double check the constants in this formula. Clearly the additive kinematic formulas commute with the globalization map glob : FlagArea The globalization of the left hand side in the kinematic formula is q k−a p a A(S k ), which equals the globalization of the right hand side by (8).