CROFTON MEASURES FOR HOLMES-THOMPSON VOLUMES IN MINKOWSKI SPACE LOUIS YANG LIU

Abstract. The space of geodesics in Minkowski space has a symplectic structure which is induced by the projection from the sphere-bundle. In this article, we show that it can be also obtained from the symplectic structure on the tangent bundle of the Riemannian manifold, the tangent bundle of the Minkowski unit sphere.

We give detailed descriptions and expositions on

Holmes-Thompson volumes in Minkowski space by the symplectic structure and the Crofton measures for them, based on which convex valuation theory and algebraic structure of the space generate by Holmes-Thompson valuations for Minkowski space can be established.

1. Introductions 1.1. Minkowski Space and Geodesics. A Minkowski space is a vector space with a Minkowski norm, and a Minkowski norm is defined in [CS] as Definition 1.1. A function F : Rn → R is a Minkowski norm if (1) F (x) > 0 for any x ∈ Rn \ {0} and F (0) = 0. (2) F (λx) = |λ|F (x) for any x ∈ Rn \ {0}. (3) F ∈ C ∞ (Rn \ {0}) and the symmetric bilinear form gx (u, v) :=

1 ∂2 2 F (x + su + tv)|s=t=0 2 ∂s∂t

(1.1)

is positively definite on Rn for any x ∈ Rn \ {0}. We denote a Minkowski space by (Rn , F ). By the way, (2) and (3) in Definition 1.1 imply the the convexity of F , see Chapter 1 of [CS]. Date: July 30, 2008. 1

2

LOUIS YANG LIU

First of all, we can infer the following theorem about geodesics in Minkowski space from Definition 1.1.

Theorem 1.2. The straight line joining two points in Minkowski space is the only shortest curve joining them.

Proof. For any p, q ∈ (Rn , F ), let r(t), t ∈ [a, b] with F (r0 (t)) = 1, be a curve joining p and q, which has the minimum length. Then r(t) is the minimizer of the ´b functional a F (r0 (t))dt. Note that F is smooth. By the fundamental lemma of calculus of variation (Let ´b V (h) := a F (r0 (t) + hδ 0 (t))dt, where δ(a) = δ(b) = 0. Then V 0 (0)

= =

´b ∂ ∂h |h=0 a

F (r0 (t) + hδ 0 (t))dt

´b

∂ | F (r0 (t) a ∂h h=0

+ hδ 0 (t))dt

´b

∇F (r0 (t)) · δ 0 (t)dt ´b = ∇F (r0 (t)) · δ(t)|ba − a δ(t) · ´b d = − a δ(t) · dt ∇F (r0 (t))dt. =

Thus we can obtain

(1.2)

a

d 0 dt ∇F (r (t))

d 0 dt ∇F (r (t))dt

= 0 since V 0 (0) = 0 as V (0) 6 V (h) for any δ(t)),

or by the Euler–Lagrange equation directly, we have d ∇F (r0 (t)) = 0. dt

(1.3)

Using chain rule, (1.3) becomes Hess(F )

d2 r(t) = 0. dt2

(1.4)

d2 r(t) =0 dt2

(1.5)

On the other hand, we have ∇F (r0 (t))

CROFTON MEASURES FOR HOLMES-THOMPSON VOLUMES IN MINKOWSKI SPACE

3

by differentating F (r0 (t)) = 1, and then by product rule, (1.4) and (1.5), 2 1 2 d r(t) 2 Hess(F ) dt2

2

2

= F (r0 (t))Hess(F ) d dtr(t) + (∇F (r0 (t))T ∇F (r0 (t)) d dtr(t) 2 2 2

= Hess(F ) d dtr(t) 2 =

0. (1.6)

2

Hence we get

d r(t) dt2

= 0 because 21 Hess(F 2 ) is non-degenerated by (3) in Definition

1.1, and then it implies r(t), t ∈ [a, b], is a straight line segment connecting p and q.

 Thus the space of geodesics in (Rn , F ) actually is the space of affine lines, denoted

by Gr1 (Rn ). More generally, one can define Definition 1.3. The affine Grassmannian Grk (Rn ) is the space of affine k-planes in (Rn , F ). 1.2. Symplectic Structures on Cotangent Bundle. The Minkowski space (Rn , F ), as a differentiable manifold, has a canonical symplectic structure on its cotangent bundle T ∗ Rn , from which a symplectic structure on its tangent bundle T Rn can be derived as well. The canonical contact form α on T ∗ Rn is defined as αξ (X) := ξ(π0∗ X) for X ∈ Tξ T ∗ Rn , where π0 : T ∗ Rn → Rn is the natural projection. And then the canonical symplectic form on T ∗ Rn is defined as ω := dα. On the other hand, we know that the dual of Minkowski metric is defined as F ∗ (ξ) := sup {|ξ(v)| : v ∈ T Rn , F (v) 6 1}

(1.7)

for ξ ∈ T ∗ Rn , and there is a natural correspondence between the sphere bundle SRn and the cosphere bundle SRn = {ξ ∈ T ∗ Rn : F ∗ (ξ) = 1} of the Minkowski space (Rn , F ). By the convexity and the positive homogeneity of F , see [S1], we can obtain ¯ = 1 and dF ∗ (dF (ξ)) ¯ = ξ¯ for any ξ¯ ∈ Sx Rn and x ∈ Rn , where dF is the F ∗ (dF (ξ)) gradient of F and similarly for dF ∗ . Thus dF is a diffeomorphism from Sx Rn to

4

LOUIS YANG LIU

Sx∗ Rn , which induces another diffeomorphism ϕF : SRn → S ∗ Rn

(1.8)

ϕF ((x, ξ x )) = (x, dF (ξ x )) for any ξ x ∈ Sx Rn . More generally, there is another diffeomorphism

1 2 2 dF

from

Tx Rn \ {0} to Tx∗ Rn \ {0} for any x ∈ (Rn , F ), thus we obtain a diffeomorphism ϕ¯F : T Rn → T ∗ Rn

(1.9)

ϕ¯F ((x, ξ x )) = (x, 12 dF 2 (ξ x )) by ignoring the 0-sections. The diffeomorphism (1.8) induces a 2-form ω ¯ := ϕ∗F ω on SRn . Without loss of elegance, we can express it more concretely. Since T ∗ Rn = Rn × Rn∗ , for (x, ξ) ∈ T ∗ Rn the canonical symplectic form ω on T ∗ Rn is actually ω = tr(dx ∧ dξ), here we denote dx ∧ dξ := (dxi ∧ dξj )n×n and similarly dx ∧ dξ¯ := (dxi ∧ dξ¯j )n×n , ∂ ¯ = dxj (ξ) ¯ and n × n matrices with 2-forms as entries, where ξj (ξ) = ξ( ∂x ), ξ¯j (ξ) j

¯ = 1. Then using chain rule, we can obtain F ∗ (ξ)

¯ SRn , ω ¯ = ϕ∗F (ω|S ∗ Rn ) = Hess(F ) ? dx ∧ dξ|

(1.10)

where ? is the Frobenius inner product which is the sum of the entries of the entrywise product of two matrices.

1.3. Gelfand Transform. Gelfand transform on a double fibration as a generalization of Radon transform plays an important role in making use of the symplectic form of Section 1.2 in integral geometry of Minkowski space. π

π

Definition 1.4. Let M ←1 F →2 Γ be double fibration where M and Γ are two manifolds, π1 : F → M and π2 : F → Γ are two fibre bundles, and π1 × π2 : F → M × Γ is an submersion. Let Φ be a density on Γ, then the Gelfand transform of Φ is defined as GT (Φ) := π1∗ π2∗ Φ. In the case Φ is a differential form and the fibres are oriented, then we also have a well-defined Gelfand transform GT (Φ) := π1∗ π2∗ Φ,

CROFTON MEASURES FOR HOLMES-THOMPSON VOLUMES IN MINKOWSKI SPACE

5

noting that the pushforward of a form is the integral of contracted form over the fibre. To make it clear, let’s see how the degree of a density or form changes by the transform. Suppose Φ is a density or form of degree m on Γ and the dimension of ´ fibre π1 is q, then π2∗ Φ has degree m, and then GT (Φ) = π1∗ π2∗ Φ = π−1 (x) π2∗ Φ for 1

x ∈ M has degree m − q. An application of Gelfand transforms in integral geometry is the following fundamental theorem [AF1], whose proof is quite simple. Theorem 1.5. Suppose Mγ := π1 (π2−1 (γ)) are smooth submanifolds of M for γ ∈ Γ, M ⊂ M is a immersed submanifold, and Φ is a top degree density on Γ. Then

ˆ

ˆ #(M ∩ Mγ )Φ(γ) =

GT (Φ).

(1.11)

M

Γ

Proof. Working on the transitions of measures on manifolds and the transformations of intersection numbers, we have ´ M

GT (Φ)

=

´ M

π1∗ π2∗ Φ

= = =

´ ´ ´

π1−1 (M )

π2∗ Φ

Γ

#(π2−1 (γ) ∩ π1−1 (M ))Φ(γ)

Γ

#(M ∩ Mγ )Φ(γ).

(1.12)



2. The Symplectic Structure on the Space of Geodesics The symplectic structure on the space of geodesics in a Minkowski space is induced naturally from the canonical symplectic structure on its cotangent bundle. The process of construction of symplectic form on Gr1 (Rn ) in Minkowski space (Rn , F ) is based on the following diagram ϕF

SRn ↓p Gr1 (Rn )

'

→

S ∗ Rn

i

,→ T ∗ Rn (2.1)

6

LOUIS YANG LIU

where p is the projection from SRn onto Gr1 (Rn ) defined by

¯ := l(x, ξ), ¯ p((x, ξ))

(2.2)

¯ is the line passing through x with direction ξ. ¯ where l(x, ξ) Consider the geodesic vector field X (ξ x ) := (ξ x , 0) on T Rn for any ξ x ∈ SRn , ϕF in (1.8) induces another vector field X := dϕF (X ) on T ∗ Rn with X (dF (ξ x )) = (dϕF (X )(ϕF (ξ x )) = (ξ x , 0)

(2.3)

for ξ x ∈ SRn . We have the following vanishing property about X and ω on S ∗ Rn . Lemma 2.1. iX ω = 0 on S ∗ Rn . Proof. Noting that ω(X, Y ) = hX1 , Y2 i − hY1 , X2 i for any X = (X1 , X2 ) and Y = (Y1 , Y2 ) in Tξx S ∗ Rn ⊂ Tξx T ∗ Rn because T ∗ Rn ∼ = Rn ×Rn∗ , where the inner product is the dual space action, by (2.3) we have ωξx (X , Y ) = hξ x , Y2 i = hdF ∗ (ξx ), Y2 i = 0

(2.4)

because Y2 ∈ Tξx S ∗ Rn is “normal” to dF ∗ (ξx ), precisely, that can be obtained by differentiating F ∗ (ξx ) = 1 and noting Y2 ∈ Tξx S ∗ Rn .



Furthermore, the Lie derivative of ω along geodesic vector field X is LX ω = diX ω + iX dω = 0

(2.5)

by Lemma 2.1. Then (2.5) implies (ϕF )∗ ω|S ∗ Rn is invariant under X¯ . Based on the invariance of ω we can construct a symplectic structure on Gr1 (Rn ). However, in order to do that, we need to give a manifold structure for Gr1 (Rn ) first. In fact, we can build a bijection ψ between Gr1 (Rn ) and T SFn−1 ,where SFn−1 is ¯ ∈ Gr1 (Rn ), let η¯ be the tangent vector the unit sphere in (Rn , F ). For any l(x, ξ) ¯ ∩ T ¯S n−1 , in fact, η¯ = x − dF (ξ)(x) ¯ pointing at l(x.ξ) ξ¯ ∈ Tξ¯SFn−1 , see Figure 2.1 ξ F

CROFTON MEASURES FOR HOLMES-THOMPSON VOLUMES IN MINKOWSKI SPACE

7

Figure 2.1. Gr1 (Rn ) ∼ = T SFn−1

on page 7, and one can define ¯ := (ξ,¯ ¯ η ) = (ξ, ¯ x − dF (ξ)(x) ¯ ¯ ψ(l(x, ξ)) ξ).

(2.6)

Thus we have a homeomorphism ψ from Gr1 (Rn ) to T SFn−1 , and then the manifold structure on T SFn−1 provides one for Gr1 (Rn ). Let us again consider the projection (2.2) with the manifold structure on Gr1 (Rn ), and then we can obtain the following lemma

Lemma 2.2. p∗ (X¯ ) = 0, i.e. X¯ is in the kernel of dp.

Proof. Using the basic equality ¯ ξ) ¯ = F (ξ) ¯ =1 dF (ξ)(

(2.7)

8

LOUIS YANG LIU

obtained by the positive homogeneity of F for any ξ¯ ∈ Sx Rn , we have p∗ (X¯ )

¯ 0)) = dp((ξ,

¯ x − dF (ξ)(x) ¯ ¯ ξ, ¯ 0)) = d(ξ, ξ)(( ¯ ξ) ¯ ξ¯ = ξ¯ − dF (ξ)( =

¯ ξ)) ¯ ξ¯ (1 − dF (ξ)(

=

0.

(2.8)

 One can compute the rank of the Jacobian of p which is 2n − 2, that implies dim(dp|ξ¯x ) = 1 and then ker(dp|ξ¯x ) = span(X¯ (ξ¯x ))

(2.9)

by Lemma 2.2. Now we can obtain the following theorem Theorem 2.3. There exists a symplectic form ω0 on Gr1 (Rn ), such that p∗ ω0 = ω ¯ = (ϕF )∗ ω|S ∗ Rn . Proof. By (2.5), (2.9) and Lemma 2.1, we know that ω ¯ ξx (X, Y ) is independent of the choices of preimages under the pushforward induced by projection p. Thus we have a well-defined two form ω0 on Gr1 (Rn ), ˜ Y˜ ) := ω ω0p(ξx ) (X, ¯ ξx (X, Y ),

(2.10)

˜ and (p∗ )ξ (Y ) = Y˜ , such that where (p∗ )ξx (X) = X x p∗ ω 0 = ω ¯ = (ϕF )∗ i∗ ω.

(2.11) 

That finishes the construction of symplectic structure on the space of geodesics in Minkowski space. On the other hand, since T ∗ SFn−1 as a cotangent bundle on Riemannian manifold SFn−1 has a canonical symplectic structure denoted as ω ˜ , and we have a canonical

CROFTON MEASURES FOR HOLMES-THOMPSON VOLUMES IN MINKOWSKI SPACE

9

diffeomorphism ϕ˜F : T SFn−1 → T ∗ SFn−1

(2.12)

ϕ˜F (¯ ηξ¯) = h¯ ηξ¯, ·igF , in which gF is the Riemannian metric on SFn−1 , which is actually the bilinear form h¯ u, v¯igF :=

∂2 F (ξ¯ + s¯ u + t¯ v )|s=t=0 ∂s∂t

(2.13)

for any u ¯, v¯ ∈ Tξ¯SFn−1 , see [CS], and then ϕ˜∗F ω ˜ is the the symplectic form induced on T SFn−1 ∼ = Gr1 (Rn ). Also, we have another symplectic form ω0 on Gr1 (Rn ) from Theorem 2.3. A natural question is whether the two symplectic structures on Gr1 (Rn ) are the same, the answer is yes, see the following theorem ¯. Theorem 2.4. ω0 = ϕ˜∗F ω Let us first draw a diagram for this theorem by combining (2.1) ϕF

'

SRn

→

S ∗ Rn

i

,→ T ∗ Rn (2.14)

↓p ψ

Gr1 (Rn ) ' T SFn−1

ϕ ˜F

'

→

T ∗ SFn−1 .

Proof. First, differentiating (2.7) and using chain rule, one can get ¯ ξ| ¯ SRn = 0 Hess(F ) ? ξd

(2.15)

¯ ξ¯ := (ξ¯i dξ¯j )n×n is a matrix and ? is the Frobenius inner product of in which ξd matrices. Next, the canonical symplectic form ω ˜ on T ∗ SFn−1 , ω ˜ = ω|T ∗ S n−1 in which ω is F

the canonical symplectic form on the cotangent bundle T ∗ Rn . Thus, from (2.12) and (2.13), one can obtain that ¯ n−1 , ϕ˜∗F ω ˜ = Hess(F ) ? d¯ η ∧ dξ| TS

(2.16)

F

here dξ¯ ∧ d¯ η is a matrix with 2-form entries and ? is the Frobenius inner product of matrices.

10

LOUIS YANG LIU

Therefore, by plugging (2.6) into (2.16) and using (2.15), we obtain p∗ ϕ˜∗F ω ˜

¯ n−1 = Hess(F ) ? d¯ η ∧ dξ| TS F

¯ ¯ ∧ dξ| ¯ SRn = Hess(F ) ? d(x − dF (ξ)(x) ξ) ¯ SRn − d(dF (ξ)(x)) ¯ ¯ ξ| ¯ SRn = Hess(F ) ? dx ∧ dξ| ∧ Hess(F ) ? ξd

(2.17)

¯ SRn = Hess(F ) ? dx ∧ dξ| = ω ¯, which by Theorem 2.3 implies the claim.



At the end to this section, we make a remark on the symplectic structure on Gr1 (Rn ). Remark 2.5. From (1.10) we see the symplectic structure ω ¯ on T Rn relies on the Minkowski metric F , then we know, by the above construction, the symplectic structure on Gr1 (Rn ) depends on the Minkowski metric F as well. Let us see the following example of Minkowski plane with p-norm as a Minkowski metric.
Example 2.6. Given a Minkowski plane by R2 , || · ||p , 1 < p < ∞, where p ||(α, β)||p = (|α|p + |β|p )1/p and the dual norm is || · || p−1 we can obtain the sym
plectic form ω on Gr1 (R2 ), the space of affine lines in R2 , || · ||p , by following the

general construction above. By (1.10) and Theorem 2.3, we have p∗ ω0 = (p − 1)αp−2 dx ∧ dα + (p − 1)β p−2 dy ∧ dβ,

(2.18)

for ((x, y), (α, β)) ∈ SR2 . Since Gr1 (R2 ) is a 2-dimensional manifold, we can parametrize affine lines in Gr1 (R2 ) with two variables in a natural way. For any straight line l passing through (x, y) with direction (α, β) of unit p-norm, let (−Θ, Ω) be the unit vector in p-norm such that l is tangent to the Minkowski sphere S(r) of radius r at (−rΘ, rΩ), here we can call r the “p-norm distance” of l to the origin, see Figure 2.2 on page 11. Thus we can denote the line by l(r, Θ).

CROFTON MEASURES FOR HOLMES-THOMPSON VOLUMES IN MINKOWSKI SPACE 11

Figure 2.2. “p-norm distance” r

We have the following theorem about the symplectic structure on Gr1 (R2 ) by the above parametrization.

Theorem 2.7. The symplectic structure on Gr1 (R2 ) is 2

ω0 =

(p − 1)2 Θp(p−2) Ωp

−3p+1

dr ∧ dΘ.

(p−1)(2p−1)

||(Θ, Ω)||p(p−1)

(2.19)

Proof. For a line l passing through (x, y) with direction (α, β) of unit p-norm, the “p-norm distance” r = −Θp−1 x + Ωp−1 y

(2.20)

and 1

1

β p−1

(−Θ, Ω) = (− (α

p p−1

+β

p p−1

1

,

) p (α

α p−1 p p−1

p

1

+ β p−1 ) p

).

(2.21)

12

LOUIS YANG LIU

In order to express ω0 in terms of r and Θ, at first we use (2.20) and (2.21) to compute dr ∧ dΘ

= (−Θp−1 dx + Ωp−1 dy) ∧ dΘ 1

= −Θp−1 dx ∧ d( = −Θp−1 dx ∧ d(

1

β p−1 p (α p−1

+β

p p−1

1 )p

1

p

) + Ωp−1 dy ∧ d(

) + Ωp−1 dy ∧ d(

1

p−1 +1) p (( α β)

p

p+1 p

1 p βdα−αdβ α p−1 p−1 ( β ) β2

p

1 p−1 (( α ) p−1 + 1)− ( p−1 )( α β) β

1 p−1 (( α ) p−1 + 1)− ( p−1 )( α β) β p−1

β −Ωp−1 (− β1 α p−1 −

= =

1 ( p−1 )( α β)

1 p−1

1 α (p−1)2 ( β )

(( α β)

1 p−1

p p−1

(( α β)

p p−1

p+1 p

(Θp−1 dx ∧ ( β1 dα −

1

)

)

α β 2 dβ)

α β 2 dβ)

p

1

1

1 p βdα−αdβ α p−1 p−1 ( β ) β2

−Ωp−1 dy ∧ ( β1 dα − =

1

p

p−1 +1) p (( α β)

p+1 p

p−1 + 1)− +Ωp−1 dy ∧ (− p1 )(( α β) 1

p

+β p−1 ) p

p

p−1 + 1)− = −Θp−1 dx ∧ (− p1 )(( α β)

=

β p−1 p (α p−1

p+1 p

α β 2 )dy

p−1

(Θ β p+1 dx ∧ dα +

− p+1 p

+ 1)

αp β p+1 )dx

∧ dα

∧ dβ)

− p+1 p

+ 1)

(Θp−1 ( β1 +

Ωp−1 αp−1 β 2 dy

∧ dβ)

p−1

Θ p−2 ( β p+1 dx ∧ dα αp−2 (p − 1)α

p−1

+ αΩp−1 β p (p − 1)β p−2 dy ∧ dβ). (2.22) Indeed, Θp−1 Ωp−1 = β p+1 αp−2 αp−1 β p

(2.23)

β Θp−1 = p−1 Ω α

(2.24)

since

by (2.21).

CROFTON MEASURES FOR HOLMES-THOMPSON VOLUMES IN MINKOWSKI SPACE 13

Therefore, using (2.23), (2.24) and ||(Θ, Ω)||p = 1, we have dr ∧ dΘ

=

p

1 1 α p−1 p−1 (( α (p−1)2 ( β ) β)

+ 1)−

p+1 p

Θp−1 β p+1 αp−2

((p − 1)αp−2 dx ∧ dα + (p − 1)β p−2 dy ∧ dβ) p

=

1 1 α p−1 p−1 (( α (p−1)2 ( β ) β)

=

1 (p−1)2

= = =

=

p+1 p

Θp−1 β p+1 αp−2 ω0

1 p−1 Θp−1 (α β) p p+1 0 p−1 +1) p β p+1 αp−2 (( α β) p−1 Ω 1 ΘΘ p+1 (p−1)2 p−1 Ω p Ωp−1 Θ (( Θ ) +1) p ( )p+1 ( )p−2 ||(Θp−1 ,Ωp−1 )||p ||(Θp−1 ,Ωp−1 )||p

ω

ΩΘ2p−1

1 (p−1)2 (

Θp−1 Ωp−1 )p+1 ( )p−2 ||(Θp−1 ,Ωp−1 )||p ||(Θp−1 ,Ωp−1 )||p 2p−1 p−1 p−1 2p−1 ΩΘ ||(Θ ,Ω )||p 1 0 (p−1)2 Θ(p−1)(p+1) Ω(p−1)(p−2)

ω0

(2.25)

ω0

ω

p−1

=

+ 1)−

p−1

||(Θ ,Ω )||2p−1 p 1 (p−1)2 Θp(p−2) Ωp2 −3p+1 (p−1)(2p−1) ||(Θ,Ω)||p(p−1) p(p−2) 2 (p−1) Θ Ωp2 −3p+1 0

ω0

ω ,

Thus we have shown (p−1)(2p−1)

dr ∧ dΘ =

||(Θ, Ω)||p(p−1)

(p − 1)2 Θp(p−2) Ωp2 −3p+1

ω0 ,

which implies (2.19) in the claim.

(2.26) 

So from (2.18) and (2.19) we see the symplectic structure on Gr1 (R2 ) is determined by the Minkowski metric || · ||p on R2 .

3. Integral Geometry on Length in Minkowski Space The length of a straight line segment in (R2 , F ) can be obtained by integrating → the canonical contact form α introduced in Section 1.2. For any x, y ∈ R2 , let − xy be the vector from x to y, and c(t) := (x +

− → t xy − → (y − x), dF ( → → )), t ∈ [0, F (xy)] F (− xy) F (− xy)

(3.1)

be a straight line segment in T ∗ R2 . By the positive homogeneity of F, one can get the useful fact that dF (

− → − → − → xy xy xy )( − ) = F( − − → → → ) = 1. F (xy) F (xy) F (xy)

(3.2)

14

LOUIS YANG LIU

Therefore, ´ c

α

=

→ ´ F (− xy) 0

− → − → xy − → dF ( F (xy − → )( F (− → )dt = F (xy) xy) xy)

= L(xy),

(3.3)

where L(xy) is the length of xy. Here let us introduce a general definition in integral geometry first. Definition 3.1. A Crofton measure φ for a degree k measure Φ on (Rn , F ) is a measure on Grn−k (Rn ) (Definition 1.3), such that it satisfies the Crofton-type formula

ˆ #(M ∩ P )Φ(P )

Φ(M ) =

(3.4)

P ∈Grn−k (Rn )

for any compact convex subset M (Rn , F ). Furthermore, we have the following Proposition 3.2. The Crofton measure on Gr1 (R2 ) for the length is |ω0 |. Our treatment of applying Stokes’ theorem here is primarily based on [AD]. ψ

'

Proof. From Section 2, we know Gr1 (R2 ) → T SF which is a cylinder, and it has a symplectic form ω0 as T SF embedded in T ∗ R2 .   ◦ Let S := l ∈ Gr1 (R2 ) : l ∩ xy 6= φ , Cx and Cy be the family of oriented lines + − passing through x and y respectively, then Cx ∩ Cy = lxy , lxy that are the two oriented lines connecting x and y, and ∂S = Cx ∪ Cy .  Let R := ξ ∈ SR2 ⊂ T R2 : l((x + t(y − x), dF (ξ)) ∩ xy 6= φ , where l((x+t(y − x), dF (ξ)) is the line passing through x + t(y − x) with direction ξ, then p(R) = S, where p is the natural projection from SR2 to Gr1 (R2 ).   0+ 0 0 Additionally, let Cx = ξx : ξx ∈ Sx R2 , Cy = ξy : ξy ∈ Sy R2 , lxy = 0

− Sx R2 , and lxy =

− → − F (xy − → xy)

0

0

0

− → xy → F (− xy)

∈

0

+ − + ∈ Sy R2 , then p maps Cx , Cy , lxy and lxy to Cx , Cy , lxy

− and lxy respectively, see Figure 3.1 on page 15.

Applying Stokes’ theorem to the two regions individually, using the fact that ´

0

C0

0

α = 0 because of the fixed base points for any C 0 ⊂ Cx or Cy , and combining

CROFTON MEASURES FOR HOLMES-THOMPSON VOLUMES IN MINKOWSKI SPACE 15

Figure 3.1. From SR2 to Gr1 (R2 ) with (3.3), we obtain ´ S

|ω0 | =

´ p(R)

|ω0 | = =

´ ´

R

|p∗ ω0 |

R

|ω|

´

=

2

=

4L(xy).

0+ 0− ∪lxy lxy

(3.5) α

Therefore, for any rectifiable curve γ in (R2 , F ), the length of γ, 1 L(γ) = 4 which is the desired claim.

ˆ #(γ ∩ l)|ω0 |,

(3.6)

l∈Gr1 (R2 )



Remark 3.3. The proof above can be applied to R2 with projective Finsler metric, in which geodesics are straight lines. Furthermore, for Rn with a projective Finsler

16

LOUIS YANG LIU

metric F , we choose a plane P ⊂ Rn containing xy for any x, y ∈ Rn , then L(xy) =

1 4

ˆ #(xy ∩ l)|ω0 |.

(3.7)

l∈Gr1 (P )

4. Volume of Hypersurfaces A standard definition of Holmes-Thompson volume in Minkowski space (Rn , F ) is given and its importance in Finsler geometry and integral geometry is illustrated in [AB]. The Holmes-Thompson volumes are defined as follows. Definition 4.1. Let N be a k-dimensional manifold and D∗ N := {ξx ∈ T ∗ N : F ∗ (ξx ) 6 1} ,

(4.1)

where F ∗ is the dual norm in (1.7), be the codisc bundle of N , then the k-th Holmes-Thompson volume is defined as 1 volk (N ) := k

ˆ |ω k |,

(4.2)

D∗ N

where k is the Euclidean volume of k-dimensional Euclidean ball and ω is the canonical symplectic form on the cotangent bundle of N . Let Λ ∈ Grk (Rn ) for some k ≤ n, ω0 and ω ˆ 0 are the natural symplectic forms on Gr1 (Rn ) and Gr1 (Λ) constructed in the way described in Section 2. The relation between ω0 and ω ˆ 0 is shown in the following Lemma 4.2. i∗ ω0 = ω for i : Gr1 (Λ) ,→ Gr1 (Rn ). Proof. First consider the diagram ϕF ∗

'

ˆi

ϕF

'

S ∗ Λ → SΛ ,→ SRn → S ∗ Rn .

(4.3)

We have a canonical contact form α ˆ ξ (X) := ξ(ˆ π0∗ X) for X ∈ Tξ S ∗ Λ on S ∗ Λ in diagram (4.3), where π ˆ0 : S ∗ Λ → Λ is the natural projection, and define ω ˆ := dˆ α on S ∗ Λ.

CROFTON MEASURES FOR HOLMES-THOMPSON VOLUMES IN MINKOWSKI SPACE 17

Let j = ϕF ◦ ˆi ◦ ϕF ∗ , then for any X ∈ Tξ S ∗ Λ, (j ∗ α)ξ (X) = αj(ξ) (j∗ X) = j(ξ)(π∗ j∗ X) = ξ(ˆ π0∗ X) = α ˆ ξ (X)

(4.4)

in which α and ω on S ∗ Rn are introduced in Section 1.2, then (4.4) implies j∗α = α ˆ,

(4.5)

j∗ω = ω ˆ

(4.6)

and furthermore we have

by differentiating (4.5). Next, let pˆ be the projection taking ξ¯x ∈ SΛ to the line passing x with the direction ξ¯x , and similarly for p which is described in (2.1). Consider the diagram S∗Λ

ϕF ∗

'

SΛ

ˆi

,→

↓ pˆ Gr1 (Λ)

SRn

ϕF

'

S ∗ Rn (4.7)

↓p i

,→ Gr1 (Rn )

obtained by combining diagram (2.1) and (4.4). By the definitions of the maps in (4.7), we know the diagram is commutative. By Theorem 2.3, we have p∗ ω0 = ϕ∗F ω ˆ . Combining with (4.6) and the commutativity of the diagram and pˆ∗ ω ˆ 0 = ϕ∗F ω (4.7), we obtain the desired claim i∗ ω0 = ω ˆ0.



Suppose N is a hypersurface in (Rn , F ), then we have the following Proposition 4.3. voln−1 (N ) =

1 2n−1

´ l∈Gr1 (Rn )

#(N ∩ l)|ω0n−1 |, where ω0 is the

symplectic form on Gr1 (Rn ). This idea of intrinsic proof is given by Dr. Joseph H. G. Fu. Proof. It suffices to prove the claim in the case when N is affine. Without loss of generality, assume N ⊂ Rn−1 ⊂ Rn is compact and convex with smooth boundary. Consider the following diagram ˆi

ϕF ∗

∼ =

i

ϕF

∼ =

π

S ∗ N ,→ S ∗ Rn−1 → SRn−1 ,→ SRn → S ∗ Rn → Gr1 (Rn ),

(4.8)

18

LOUIS YANG LIU

where i and k are embeddings, and π := p ◦ ϕ−1 F = p ◦ ϕF ∗ is a projection from diagram (2.1). As N is a (n − 1)-dimensional manifold, the canonical contact form α ˆ on S ∗ N is defined as α ˆ ξ (X) := ξ(ˆ π0∗ X) for X ∈ Tξ S ∗ N , where π ˆ0 : S ∗ N → N is the projection. Let j = ϕF ◦ i ◦ ϕF ∗ , then (j ∗ α)ˆi(ξ) (ˆi∗ X) = ((ϕF ◦ i ◦ ϕF ∗ )∗ α)ˆi(ξ) (ˆi∗ X) = ξ(π0∗ X) = α ˆ ξ (X).

(4.9)

for any X ∈ Tξ S ∗ N , which implies (ˆi ◦ j)∗ α = ˆi∗ j ∗ α = α ˆ , and then (i ◦ j)∗ ω = ω ˆ where ω ˆ := dˆ α and ω is introduced in Section 1.2. Applying Stokes’ theorem, we have ´ D∗ N

ω ˆ n−1

=

´ ∂(D ∗ N )

α ˆ∧ω ˆ n−2

= =

´ ´

S∗N

α ˆ∧ω ˆ n−2 +

S∗N

α ˆ∧ω ˆ n−2

´ π ˆ 0−1 (∂N )

α ˆ∧ω ˆ n−2

(4.10) since the degree of α ˆ∧ω ˆ n−2 on the compoment mesuring perturbations of base points is bigger than the dimension of the base manifold, and ´ ∗ Rn ∩π −1 (N ) S+ 0

ω n−1

´

α ∧ ω n−2 ´ ´ = S ∗ N ˆi∗ j ∗ α ∧ ˆi∗ j ∗ ω n−2 + πˆ −1 (∂N ) ˆi∗ j ∗ α ∧ ˆi∗ j ∗ ω n−2 0 ´ n−2 = S∗N α ˆ∧ω ˆ , (4.11)

=

∗ Rn ∩π −1 (N )) ∂(S+ 0

where  ∗ n S+ R = ξ ∈ S ∗ Rn : ξ(v0 ) > 0, v0 satisfies dF (v0 )(v) = 0 for all v ∈ SRn−1 . (4.12) Therefore,

ˆ

ˆ ω ˆ n−1 =

D∗ N

ω n−1 . ∗ Rn ∩π −1 (N ) S+ 2

(4.13)

CROFTON MEASURES FOR HOLMES-THOMPSON VOLUMES IN MINKOWSKI SPACE 19

Now let us consider the “upper” half space of geodesics in (Rn , F ),  Gr1+ (Rn ) := l(x, η) : dF (η)(η0 ) > 0, η0 satisfies dF (η0 )(v) = 0 for all v ∈ SRn−1 . (4.14) Since π ∗ ω0 = ω, then we get ´ ∗ Rn ∩π −1 (N ) S+ 0

ω n−1

= = =

´ ´ ´

∗ Rn ∩π −1 (N ) S+ 0

π ∗ ω0n−1

∗ Rn ∩π −1 (N ) π −1 (l)∈S+ 0

l∈Gr1+ (Rn )

#(N ∩ l)ω0n−1

(4.15)

#(N ∩ l)ω0n−1 .

Combining with (4.13), we obtain voln−1 (N )

´

=

1 n−1

=

1 n−1

=

1 2n−1

´

D∗ N

ω ˆ n−1

l∈Gr1+ (Rn )

´

l∈Gr1 (Rn )

#(N ∩ l)ω0n−1

(4.16)

#(N ∩ l)|ω0n−1 |, 

that finishes the proof.

5. k-th Holmes-Thompson Volume and Crofton Measures Let us introduce a general fact first. Busemann constructed all projective metrics F for projective Finsler space (Rn , F ), and it was also proved in [S2] by Schneider using spherical harmonics.

Theorem 5.1. (Busemann) Suppose F is a projective metric on Rn , then F (x, v) = ´ |hξ, vi|f (ξ, hξ, xi)Ω0 for any (x, v) ∈ T Rn , where Ω0 is the Euclidean volume ξ∈S n−1 form on S n−1 and f is some continuous function on S n−1 × R. In fact, for the case that (Rn , F ) is Minkowski, we can use a theorem on surjectivity of cosine transform, ˆ C(f )(·) =

|hξ, ·i|f (ξ)Ω0 , ξ∈S n−1

of even functions from Chapter 3 of [G],

(5.1)

20

LOUIS YANG LIU

Theorem 5.2. For any even C 2[(n+3)/2] function g on S n−1 , n > 2, where [·] is the greatest integer function, there is an even function f on S n−1 such that C(f ) = g.

From it we directly obtain that there exists an even function f on S n−1 , such that 1 L(xy) = 4

ˆ

→ (ξ)Ω . |hξ, − xyi|f 0

(5.2)

ξ∈S n−1

→ x, y ∈ (R2 , F ), by Proposition 3.2 we know xy, On the other hand, for any v = − F (x, v) =

ˆ

1 ωn−1

#(xy ∩ l)|ω0 |.

(5.3)

l∈Gr1 (R2 )

In fact, there is a relation between Ω0 and ω0 . Considering the following double fibration π

π

Gr1 (Rn ) ←1 I →2 Grn−1 (Rn ), o n where I = (l, H) ∈ Gr1 (Rn ) × Grn−1 (Rn ) : l ⊂ H , we have

(5.4)

Proposition 5.3. GT (f Ω0 ∧ dr) = ω0 , where GT is the Gelfand transform for the double fibraton (5.4) and r is the Euclidean distance of a hyperplane H to the origin.

Proof. For any x, y in any 2-plane Π ⊂ Rn , we know that the length of xy, 1 L(xy) = 4

ˆ l∈Gr1 (Π)

#(xy ∩ l)|ω0 |Gr1 (Π) .

(5.5)

n o where Gr1 (Π) := l ∈ Gr1 (Rn ) : l ⊂ Π . n o Let I = l ∈ Gr1 (Π) ⊂ Gr1 (Rn ) : xy ∩ l 6= φ and GH := π1 (π2−1 (H)) for H ∈ Grn−1 (Rn ). By the fundamental theorem of Gelfand transform, Theorem 1.5, ˆ

ˆ #(I ∩ GH )|f Ω0 ∧ dr| =

H∈Grn−1 (Rn )

|GT (f Ω0 ∧ dr)|. I

(5.6)

CROFTON MEASURES FOR HOLMES-THOMPSON VOLUMES IN MINKOWSKI SPACE 21

Therefore, ´ l∈Gr1 (Π)

#(xy ∩ l)|GT (f Ω0 ∧ dr)| = = =

´ ´ ´

I

|GT (f Ω0 ∧ dr)|

H∈Grn−1 (Rn ) ξ∈S n−1

#(I ∩ GH )|f Ω0 ∧ dr|

→ (ξ)Ω |hξ, − xyi|f 0 (5.7)

since Grn−1 (Rn ) ∼ = S n−1 × R. By (5.2)and (5.5) we thus obtain ˆ

ˆ #(xy ∩ l)|GT (f Ω0 ∧ dr)| =

l∈Gr1 (Π)

#(xy ∩ l)|ω0 |,

(5.8)

l∈Gr1 (Π)

which implies GT (f Ω0 ∧ dr)|Gr1 (Π) = ω0 |Gr1 (Π) for any plane Π ⊂ Rn by the injectivity of cosine transform (5.1).(In Chapter 3 of [G] Groemer shows by using condensed harmonic expansion and Parseval’s equation, that C(f1 ) = C(f2 ) iff f1+ = f2+ , where f1+ (v) =

f1 (v)+f1 (−v) 2

and similarly for f2+ , for any bounded integrable

functions f1 and f2 on S n−1 .) Now define a basis for Tl Gr1 (Rn ), the tangent space of Gr1 (Rn ) at l ∈ Gr1 (Rn ). ψ

Note that Gr1 (Rn ) ' T SFn−1 from Section 2. Let {ei : i = 1, · · · , n} be the basis for Rn , and curve γ i with γ i (t) = l+tei for i = 1, · · · , n−1, where l ∈ Gr1 (Rn ), and then define ei := γ 0i (0) for i = 1, · · · , n − 1. Let l(x, ξ) be a line in Gr1 (Rn ) passing through x with direction ξ and ri (t)(ξ) for be the rotation about origin with the direction from en towards ei for time t, then let vi (t) be the parallel transport from ψ(l(x, ξ) along on ri (t)(ξ) on SFn−1 , and then define curves γ i (t) = ψ −1 (vi (t)) for  0 i = 1, · · · , n − 1, thus we can define ei := γ i (0). Then ei , ej : i, j = 1, · · · , n − 1 is a basis for Tl Gr1 (Rn ). Here we have four cases to discuss. First of all, one can obtain the fact GT (f Ω0 ∧ dr)(ei , ei ) = ω0 (ei , ei )

(5.9)

by choosing a plane Πi with the tangent space of Gr1 (Πi ) spanned by ei and ei for i = 1, · · · , n − 1.

22

LOUIS YANG LIU

On the other hand, in the double fibration (5.4), π2 |π−1 (Lij ) , in which Lij is be the 1

lines in Gr1

(Rn )

obtained by translation along ei or ej for i, j = 1, · · · , n − 1, is not

a submersion from π1−1 (Lij ) to Grn−1 (Rn ). Precisely, choose e˜i and e˜j in T(l,H) I, l ⊂ H such that dπ1 (˜ ei ) = ei and dπ1 (˜ ej ) = ej , moreover, dπ2 (˜ ei ) and dπ2 (˜ ei ) are linearly dependent in TH Grn−1 (Rn ). Therefore π1∗ π2∗ (f Ω0 ∧ dr)l (ei , ej ) = ´ π ∗ (f Ω0 ∧ dr)(ei , ej ) = 0 for i, j = 1, · · · , n − 1, and obviously ω0 (ei , ej ) = 0, π −1 (l) 2 1

thus GT (f Ω0 ∧ dr)(ei , ej ) = ω0 (ei , ej ) = 0

(5.10)

for i, j = 1, · · · , n − 1. ¯ ij be the lines in For the case of ei and ej , i 6= j, i = 1, · · · , n − 1. Let L Gr1 (Rn ) obtained by translation along ei or rotation along ej . Again, π2 |L¯ ij in ¯ ij ) to Grn−1 (Rn ) either, and it also can be (5.4) is not a submersion from π1−1 (L explained precisely as the above case, therefore π1∗ π2∗ (f Ω0 ∧ dr)(ei , ej ) = 0 for i, j = 1, · · · , n − 1, and obviously ω0 (ei , ej ) = 0, thus GT (f Ω0 ∧ dr)(ei , ej ) = ω0 (ei , ej ) = 0

(5.11)

for i 6= j, i, j = 1, · · · , n − 1. Similarly for the last case of ei and ej , i, j = 1, · · · , n − 1, GT (f Ω0 ∧ dr)(ei , ej ) = ω0 (ei , ej ) = 0. So we have GT (f Ω0 ∧ dr) = ω0 on Gr1 (Rn ).

(5.12) 

One can use the diagonal intersection map and Gelfand transform by following [AF2] to construct Crofton measure for the k-th Holmes-Thompson volume. Let Ωn−1 := f Ω0 ∧ dr and define a map k

π : Grn−1 (Rn ) \4k → Grn−k (Rn )

(5.13)

π((H1 , · · · , Hk )) = H1 ∩ · · · ∩ Hk , where 4k = {(H1 , · · · , Hk ) : dim(H1 ∩ · · · ∩ Hk ) > n − k} and then let Ωn−k := π∗ Ωkn−1 .

CROFTON MEASURES FOR HOLMES-THOMPSON VOLUMES IN MINKOWSKI SPACE 23

Now consider the following double fibration, π1,k

π2,k

Gr1 (Rn ) ← Ik → Grn−k (Rn ),

(5.14)

o n where Ik = (l, S) ∈ Gr1 (Rn ) × Grn−k (Rn ) : l ⊂ S . Then we have the following proposition about the Gelfand transform on (5.14)

Proposition 5.4. GT (Ωn−k ) = ω0k .

Proof. Let n o k H := (l, (H1 , H2 , · · · , Hk )) ∈ Gr1 (Rn ) × Grn−1 (Rn ) : l ⊂ H1 ∩ · · · ∩ Hk (5.15) and consider the following diagram Gr1 (Rn )

π1,k

←

Ik

π ˜1-

↑π ˜ H

π2,k

→

Grn−k (Rn ) (5.16)

↑π π ˜

→2

k

Grn−1 (Rn ) ,

in which π ˜ : H → Ik is defined by π ˜ ((l, (H1 , H2 , · · · , Hk ))) = (l, H1 ∩H2 ∩· · ·∩Hk )). Note that π1∗ π2∗ Ωn−1 = ω0 ,

(5.17)

by Proposition 5.3. For the lower part of the diagram (5.16), π ˜

π ˜

k

Gr1 (Rn ) ←1 H →2 Gr1 (Rn ) ,

(5.18)

By manipulating the map π ˜2 = π2 × · · · × π2 , the product of k copies of the map | {z } k

π2 , applying Fubini theorem for (5.17) and using the fact that π ˜1 × π ˜2 : H → k

Gr1 (Rn ) × Gr1 (Rn ) is an immersion, one can infer π ˜1∗ π ˜2∗ Ωkn−1 = ω0k . ∗ Thus, by the commutativity of the diagram (5.16) we obtain π1,k∗ π2,k Ωn−k =

ω0k .



24

LOUIS YANG LIU

In order to study the k-th Holmes-Thompson volume, one can restrict on some k + 1-dimensional flat subspace. So fix S ∈ Grk+1 (Rn ) and then define a map by intersection πS : Grn−k (Rn ) \ 4(S) → Gr1 (S)

(5.19)

πS (H n−k ) = H n−k ∩ S for H n−k ∈ Grn−k (Rn ) \ 4(S), where n o 4(S) := H n−k ∈ Grn−k (Rn ) : dim(H n−k ∩ S) > 0 .

(5.20)

Then we have the following proposition Proposition 5.5. (πS )∗ Ωn−k = ω0k |Gr1 (S) . ∗ Proof. From Proposition 5.4, we know that π1,k∗ π2,k Ωn−k = ω0k for the double π2,k

π1,k

fibration Grn−k (Rn ) ← Ik → Gr1 (Rn ). Therefore, one can obtain by the definition of the intersection map (5.19) ∗ (πS )∗ Ωn−k = π1,k∗ π2,k Ωn−k |Gr1 (S) = ω0k |Gr1 (S) .

(5.21) 

Finally, one can obtain the following theorem about Holmes-Thompson volumes. Theorem 5.6. (Alvarez) Suppose N is a k-dimensional submanifold in (Rn , F ). ´ Then volk (N ) = 21k P ∈Grn−k (Rn ) #(N ∩ P )|Ωn−k |. Proof. By Proposition 4.3, the claim is true for hypersurface case. It is sufficient to show the claim for the case when N ⊂ S for some S ∈ Grk+1 (Rn ). We obtain by Proposition 4.3 and Proposition 5.5, volk (N )

as desired.

=

1 2k

=

1 2k

=

1 2k

´ ´ ´

l∈Gr1 (S)

#(N ∩ l)|ω0k |

l∈Gr1 (S)

#(N ∩ l)|(πS )∗ Ωn−k |

P ∈Grn−k (Rn )

(5.22)

#(N ∩ P )|Ωn−k |. 

CROFTON MEASURES FOR HOLMES-THOMPSON VOLUMES IN MINKOWSKI SPACE 25

6. Valuation Theory of Holmes-Thompson Volumes From Theorem 5.6, we know the Crofton measures for k-th Holmes-Thompson volume volk is φk := |Ωn−k |. Therefore, one can extend these volumes to valuations ˆ #(K ∩ H)φk (H)

µk (K) :=

(6.1)

H∈Grn−k (Rn )

for any compact convex subset K of the Minkowski space (Rn , F ), and by Klain embedding [K] these extensions are unique. We call them Holmes-Thompson valuations. There is an important multiplication, Alesker product, see [BF], in the space of smooth convex valuations [A] on (Rn , F ). Let µA (K) := vol(K + A) and µB (K) := vol(K + B), for any convex bodies A and B with strictly convex smooth boundaries in (Rn , F ), then the Alesker product of µA,k and µB,l is defined as µA · µB (K) := vol × vol(∆(K) + A × B),

(6.2)

where ∆(K) is the diagonal embedding of K in Rn × Rn . (6.2) can be extended to the product for µk , ˆ µA · µB (K) =

µA (K ∩ (x − B)dµB (x),

(6.3)

Rn

and then the Alesker product of µk and µl , µk · µl , is defined by the extension of linear span of (6.3). In [Ber], Bernig proves by showing µk · µl (K) =

´ Grn−k (Rn )

µl (K ∩ H)dµk (H),

that the Alesker product of µk and µl equals to another product of µk and µl from Alvarez’s construction [AF2] by taking intersections of affine k-planes and l-planes in (Rn , F ), that is described as follows. Define the intersection map πk,l : Grn−k (Rn ) × Grn−l (Rn ) \ ∆k,l → Grn−k−l (Rn ), πk,l ((H1 , H2 ) = H1 ∩ H2 , where ∆k,l = {(H1 , H2 ) : dim(H1 ∩ H2 ) > n − k − l}. Then a product of µk and µl can ∗ be defined as the valuation determined by the Crofton measure πk,l (φk × φl ) as

(6.1).

26

LOUIS YANG LIU

Thus Alesker product as algebraic multiplication structure of Holmes-Thompson valuations are as follows µk · µl = µk+l .

(6.4)

Algebraically, Theorem 6.1. (Alesker-Bernig) The space of valuations generated by HolmesThompson valuations is a graded ring with respect to Alesker product. Acknowledgement. Thanks to Dr. Joseph H. G. Fu for his guidance on this subject. References [A] Semyon Alesker, Theory of Valuations on Manifolds: A Survey, Geometric And Functional Analysis, Volume 17, Number 4, November, 2007. [AB] J. C. Álvarez Paiva and Gautier Berck, What is wrong with the Hausdorff measure in Finsler spaces, Advances in Mathematics Volume 204, Issue 2, 20 August 2006, Pages 647-663. [AD] J. C. Álvarez Paiva and C.E. Durán, An introduction to Finsler geometry, Publicaciones de la Escuela Venezolana de Matématicas, Caracas, Venezuela, 1998. [AF1] J. C. Álvarez Paiva and E. Fernande, Crofton formulas and Gelfand transforms. Accept´e pour publication dans Selecta Mathematica, 21pp. [AF2] J. C. Álvarez Paiva and E. Fernande, Crofton formulas for projective Finsler spaces, Electronic Research Announcements of the AMS, 4 (1998) 91-100. [Ber] Andreas Bernig, Valuations with Crofton formula and Finsler geometry, Advances in Mathematics, 2007; 210 (2). [BF] Andreas Bernig and Joseph H. G. Fu, Convolution of convex valuations, Geometriae Dedicata, 2006. [CS] Shiing-Shen Chern, Zhongmin Shen, Riemann-Finsler Geometry, World Scientific, Published 2005. [G] H. Groemer, Geometric Applications of Fourier Series and Spherical Harmonics, Cambridge University Press, 1996. [K] Dan Klain, Even valuations on convex bodies, Transactions of the American Mathematical Society, 352 (2000), no. 1, 71-93. [S1] Rolf Schneider, Convex Bodies: The Brunn-Minkowski Theory, Cambridge University Press, 1993 [S2] Rolf Schneider, Zur einem Problem von Shephard über die Projektionen konvexer Körper (in German), Math. Z. 1967, 101: 71–82.