ON EXPLICIT HOLMES-THOMPSON AREA FORMULA IN INTEGRAL GEOMETRY LOUIS Y. LIU In this article, we give an exposition on the Holmes-Thompson theory deveoped by Alvarez. The space of geodesics in Minkowski space has a symplectic structure which is induced by the projection from the spherebundle. 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. For the Minkowski plane, a normed two dimensional space, we express the area explicitly in an integral geometry way, by putting a measure on the plane, which gives an extension of Alvarez's result for higher dimensional cases. Abstract.

1. Introductions 1.1.

Minkowski Space and Geodesics.

A Minkowski space is a vector space

with a Minkowski norm, and a Minkowski norm is dened in [8] as

Denition 1.1. (1) (2) (3)

A function

F : Rn → R

is a Minkowski norm if

F (x) > 0 for any x ∈ Rn \ {0} and F (0) = 0. F (λx) = |λ|F (x) for any x ∈ Rn \ {0}. F ∈ C ∞ (Rn \ {0}) and the symmetric bilinear

form

1 ∂2 2 F (x + su + tv)|s=t=0 (1.1) 2 ∂s∂t n n is positively denite on R for any x ∈ R \ {0}. n We denote a Minkowski space by (R , F ). By the way, (2) and (3) in Denition 1.1 imply the the convexity of F , see Chapter 1 of [8]. gx (u, v) :=

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

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

For any

joining

p

and

functional

´b a

p, q ∈ (Rn , F ),

q , which has F (r0 (t))dt.

let

r(t), t ∈ [a, b]

F (r0 (t)) = 1, be a curve r(t) is the minimizer of the

with

the minimum length. Then

Date : November 2, 2008. 2000 Mathematics Subject Classication. 28A75, 51A50, 32F17, 53B40. Key words and phrases. Minkowski space, Holmes-Thompson Volume, symplectic structure,

convex valuation.

1

2

LOUIS Y. LIU

F is smooth. By the fundamental lemma of calculus of variation (Let F (r0 (t) + hδ 0 (t))dt, where δ(a) = δ(b) = 0. Then ´b ∂ V 0 (0) = ∂h | F (r0 (t) + hδ 0 (t))dt a ´ b h=0 ∂ = a ∂h |h=0 F (r0 (t) + hδ 0 (t))dt ´b (1.2) = a ∇F (r0 (t)) · δ 0 (t)dt ´b d 0 b 0 = ∇F (r (t)) · δ(t)|a − a δ(t) · dt ∇F (r (t))dt ´b d = − a δ(t) · dt ∇F (r0 (t))dt.

Note that

V (h) :=

´b a

d 0 0 dt ∇F (r (t)) = 0 since V (0) = 0 as or by the EulerLagrange equation directly, we have Thus we can obtain

V (0) 6 V (h)

for any

δ(t)),

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)) by dierentating

F (r0 (t)) = 1,

2 1 2 d r(t) 2 Hess(F ) dt2

and then by product rule, (1.4) and (1.5), 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) d2 r(t) 1 2 Hence we get dt2 = 0 because 2 Hess(F ) is non-degenerated by (3) in Denition 1.1, and then it implies r(t), t ∈ [a, b], is a straight line segment connecting p and

q.



(Rn , F ) actually is the space of ane lines, denoted n by Gr1 (R ). More generally, one can dene Thus the space of geodesics in

Denition 1.3. in

The ane Grassmannian

Grk (Rn )

is the space of ane

k -planes

n

(R , F ).

1.2.

Symplectic Structures on Cotangent Bundle.

The Minkowski space

(Rn , F ),

as a dierentiable 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.

α on T ∗ Rn is dened as αξ (X) := ξ(π0∗ X) for π0 : T ∗ Rn → Rn is the natural projection. And then the ∗ n form on T R is dened as ω := dα.

The canonical contact form

X ∈ Tξ T ∗ Rn ,

where

canonical symplectic

On the other hand, we know that the dual of Minkowski metric is dened as

F ∗ (ξ) := sup {|ξ(v)| : v ∈ T Rn , F (v) 6 1} ∗

n

for ξ ∈ T R , and there is a natural correspondence between SRn and the cosphere bundle SRn = {ξ ∈ T ∗ Rn : F ∗ (ξ) = 1} n space (R , F ).

(1.7) the sphere bundle of the Minkowski

ON EXPLICIT HOLMES-THOMPSON AREA FORMULA IN INTEGRAL GEOMETRY

3

By the convexity and the positive homogeneity of F , see [11], we can obtain ¯ = 1 and dF ∗ (dF (ξ)) ¯ = ξ¯ for any ξ¯ ∈ Sx Rn and x ∈ Rn , where dF is the F ∗ (dF (ξ)) ∗ n gradient of F and similarly for dF . Thus dF is a dieomorphism from Sx R to ∗ n Sx R , which induces another dieomorphism

ϕF : SRn → S ∗ Rn ϕF ((x, ξ x )) = (x, dF (ξ x ))

(1.8)

ξ x ∈ Sx Rn . More generally, there is another dieomorphism 21 dF 2 from Tx R \ {0} to Tx∗ Rn \ {0} for any x ∈ (Rn , F ), thus we obtain a dieomorphism for any

n

ϕ¯F : T Rn → T ∗ Rn ϕ¯F ((x, ξ x )) = (x, 12 dF 2 (ξ x )) by ignoring the

(1.9)

0-sections.

2-form ω ¯ := ϕ∗F ω on SRn . Without loss of ∗ n n n∗ elegance, we can express it more concretely. Since T R = R × R , for (x, ξ) ∈ ∗ n ∗ n T R the canonical symplectic form ω on T R is actually ω = tr(dx ∧ dξ), here ¯ := (dxi ∧ dξ¯j )n×n , we denote dx ∧ dξ := (dxi ∧ dξj )n×n and similarly dx ∧ dξ ∂ ¯ = dxj (ξ) ¯ and n × n matrices with 2-forms as entries, where ξj (ξ) = ξ( ∂xj ), ξ¯j (ξ) ¯ = 1. Then using chain rule, we can obtain F ∗ (ξ) The dieomorphism (1.8) induces a

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

?

(1.10)

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 bration as a general-

ization of Radon transform plays an important role in making use of the symplectic form of Section 1.2 in integral geometry of Minkowski space.

Denition 1.4.

π

π

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

noting that the pushforward of a form is the integral of contracted form over the bre. To make it clear, let's see how the degree of a density or form changes by the

Φ is a density or form of degree m on Γ and the ´ dimension of 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 . transform. Suppose

bre

An application of Gelfand transforms in integral geometry is the following fundamental theorem [5], 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) =

Γ

GT (Φ). M

(1.11)

4

LOUIS Y. LIU

Proof.

Working on the transitions of measures on manifolds and the transforma-

tions of intersection numbers, we have

´

M

GT (Φ)

´

=

M

π1∗ π2∗ Φ

´ π∗ Φ −1 ) 2 ´π1 (M −1 −1 ´Γ #(π2 (γ) ∩ π1 (M ))Φ(γ) #(M ∩ Mγ )Φ(γ). 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

(Rn , F )

ϕF

SRn ↓p Gr1 (Rn ) where

p

Gr1 (Rn )

in Minkowski space

is based on the following diagram

is the projection from

'

S ∗ Rn

→

SRn

onto

i

,→ T ∗ Rn

Gr1 (Rn )

(2.1)

dened by

¯ := l(x, ξ), ¯ p((x, ξ)) (2.2) ¯ where is the line passing through x with direction ξ . n n Consider the geodesic vector eld X (ξ x ) := (ξ x , 0) on T R for any ξ x ∈ SR , ∗ n ϕF in (1.8) induces another vector eld X := dϕF (X ) on T R with ¯ l(x, ξ)

X (dF (ξ x )) = (dϕF (X )(ϕF (ξ x )) = (ξ x , 0) for

ξ x ∈ SRn . We have the following vanishing property about

Lemma 2.1.

X

and

ω

on

(2.3)

S ∗ Rn .

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 because

∗

∗

n

Y2 ∈ Tξx S R is normal to dF (ξx ), precisely, F ∗ (ξx ) = 1 and noting Y2 ∈ Tξx S ∗ Rn .

(2.4)

that can be obtained by



dierentiating

Furthermore, the Lie derivative of

ω

along geodesic vector eld

X

is

LX ω = diX ω + iX dω = 0 ∗

(2.5)

X¯ .

(ϕF ) ω|S ∗ Rn is invariant under 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 ) rst. n−1 n−1 In fact, we can build a bijection ψ between Gr1 (Rn ) and T SF ,where SF is n n ¯ the unit sphere in (R , F ). For any l(x, ξ) ∈ Gr1 (R ), let η ¯ be the tangent vector ¯ ∩ 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 by Lemma 2.1. Then (2.5) implies Based on the invariance of

on page 5, and one can dene

¯ := (ξ,¯ ¯ η ) = (ξ, ¯ x − dF (ξ)(x) ¯ ¯ ψ(l(x, ξ)) ξ).

(2.6)

ON EXPLICIT HOLMES-THOMPSON AREA FORMULA IN INTEGRAL GEOMETRY

Gr1 (Rn )

Figure 2.1.

Dieomorphic to

5

T SFn−1

ψ from Gr1 (Rn ) to T SFn−1 , and then the manifold n−1 structure on T SF provides one for Gr1 (Rn ). Let us again consider the projection (2.2) with the manifold structure on Gr1 (Rn ),

Thus we have a homeomorphism

and then we can obtain the following lemma

Lemma 2.2. Proof.

X¯ is in the kernel of dp, in other words, p∗ (X¯ ) = 0.

Using the basic equality

¯ ξ) ¯ = F (ξ) ¯ =1 dF (ξ)( obtained by the positive homogeneity of

p∗ (X¯ )

¯ 0)) = dp((ξ,

F

for any

(2.7)

ξ¯ ∈ Sx Rn ,

we have

¯ 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

dim(dp|ξ¯x ) = 1

p

which is

2n − 2,

that implies

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

ω ¯ = (ϕF ) ω|S ∗ Rn . ∗

ω0 on Gr1 (Rn ), such that p∗ ω0 =

6

LOUIS Y. LIU

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

ω0

have a well-dened two form

on

˜ Y˜ ) := ω ω0p(ξx ) (X, ¯ ξx (X, Y ), where

˜ (p∗ )ξx (X) = X

and

p.

Thus we

Gr1 (Rn ),

(p∗ )ξx (Y ) = Y˜ ,

(2.10)

such that

p∗ ω0 = ω ¯ = (ϕF )∗ i∗ ω.

(2.11)

 That nishes the construction of symplectic structure on the space of geodesics in Minkowski space.

∗ n−1 On the other hand, since T SF as a cotangent bundle on Riemannian manifold n−1 SF has a canonical symplectic structure denoted as ω ˜ , and we have a canonical dieomorphism

ϕ˜F : T SFn−1 → T ∗ SFn−1 ϕ˜F (¯ ηξ¯) = h¯ ηξ¯, ·igF , in which

gF

is the Riemannian metric on

h¯ u, v¯igF :=

SFn−1 ,

(2.12)

which is actually the bilinear form

∂2 F (ξ¯ + s¯ u + t¯ v )|s=t=0 ∂s∂t

(2.13)

˜ is the the symplectic form induced u ¯, v¯ ∈ Tξ¯SFn−1 , see [8], and then ϕ˜∗F ω n−1 ∼ on T SF = 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 for any

Theorem 2.4.

ω0 = ϕ˜∗F ω ¯.

Let us rst draw a diagram for this theorem by combining (2.1) ϕF

'

SRn ↓p

,→ T ∗ Rn (2.14)

ψ

Gr1 (Rn ) ' T SFn−1

Proof.

i

S ∗ Rn

→ ϕ ˜F

'

→

T ∗ SFn−1 .

First, dierentiating (2.7) and using chain rule, one can get

¯ ξ| ¯ SRn = 0 Hess(F ) ? ξd in which

¯ ξ¯ := (ξ¯i dξ¯j )n×n ξd

is a matrix and

?

(2.15)

is the Frobenius inner product of

matrices.

T ∗ SFn−1 , ω ˜ = ω|T ∗ S n−1 in which ω is F ∗ n cotangent bundle T R . Thus, from (2.12)

Next, the canonical symplectic form the canonical symplectic form on the

ω ˜

on

and (2.13), one can obtain that

¯ n−1 , ϕ˜∗F ω ˜ = Hess(F ) ? d¯ η ∧ dξ| TS F

here

dξ¯ ∧ d¯ η

of matrices.

is a matrix with

2-form

entries and

?

(2.16)

is the Frobenius inner product

ON EXPLICIT HOLMES-THOMPSON AREA FORMULA IN INTEGRAL GEOMETRY

Figure 2.2.  p-norm distance

7

r

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| T SF ¯ ¯ ∧ dξ| ¯ SRn Hess(F ) ? d(x − dF (ξ)(x) ξ) ¯ SRn − d(dF (ξ)(x)) ¯ ¯ ξ| ¯ SRn Hess(F ) ? dx ∧ dξ| ∧ Hess(F ) ? ξd ¯ SRn Hess(F ) ? dx ∧ dξ| ω ¯,

= = = = =

(2.17)



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

ω ¯

From (1.10) we see the symplectic structure

F, Gr1 (Rn )

on

T Rn

relies on the

Minkowski metric

then we know, by the above construction, the symplectic

structure on

depends on the Minkowski metric

following example of Minkowski plane with

Example 2.6.

Given a Minkowski plane by

||(α, β)||p = (|α|p + |β|p )1/p plectic form

ω

p-norm

on

Gr1 (R2 ),

and the dual norm

F

as well. Let us see the

as a Minkowski metric.

R2 , || · ||p is || · || p p−1

the space of ane lines in




,

1 < p < ∞,

where

we can obtain the sym-

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)

((x, y), (α, β)) ∈ SR2 . Since Gr1 (R2 ) is a 2-dimensional manifold, we can parametrize ane 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 7. Thus we can denote the line by l(r, Θ). for

8

LOUIS Y. LIU

Gr1 (R2 )

We have the following theorem about the symplectic structure on

by

the above parametrization.

Theorem 2.7. The symplectic structure on Gr1 (R2 ) is ω0 =

Proof.

For a line

l

(p − 1)2 Θp(p−2) Ωp

2

−3p+1

dr ∧ dΘ.

(p−1)(2p−1) ||(Θ, Ω)||p(p−1)

passing through

(x, y)

with direction

(2.19)

(α, β)

p-norm,

of unit

the

 p-norm distance

r = −Θp−1 x + Ωp−1 y

(2.20)

and 1

1

β p−1

(−Θ, Ω) = (−

p

p

1

α p−1

,

p

p

1

).

(2.21)

(α p−1 + β p−1 ) p (α p−1 + β p−1 ) p In order to express

ω0

r

in terms of

and

Θ,

at rst we use (2.20) and (2.21) to

compute

dr ∧ dΘ

=

(−Θp−1 dx + Ωp−1 dy) ∧ dΘ 1

= −Θp−1 dx ∧ d( = −Θ = = = = =

p−1

dx ∧ d(

1

β p−1 p (α p−1

+β 1

p

p p−1 1

1 )p

) + Ωp−1 dy ∧ d(

)+Ω

p−1

dy ∧ d(

β p−1 p (α p−1

p

p

1

)

+β p−1 ) p

1 1

)

p−1 +1) p p−1 +1) p (( α (( α β) β) p 1 p βdα−αdβ − p+1 1 α p−1 α p−1 p−1 p + 1) −Θ dx ∧ (− p )(( β ) p−1 ( β ) β2 p 1 p βdα−αdβ − p+1 1 α p−1 α p−1 p−1 p +Ω dy ∧ (− p )(( β ) + 1) p−1 ( β ) β2 p 1 − p+1 α p−1 α p−1 1 1 p−1 p (( β ) + 1) (Θ dx ∧ ( β dα − βα2 dβ) ( p−1 )( β ) 1 α p−1 −Ω dy ∧ ( β dα − β 2 dβ) p p+1 1 1 αp p−1 (( α ) p−1 + 1)− p (Θp−1 ( 1 + ( p−1 )( α β) β β β p+1 )dx ∧ dα β p−1 α −Ωp−1 (− β1 α p−1 − β 2 )dy ∧ dβ) p+1 p 1 p−1 1 Ωp−1 p−1 (( α ) p−1 + 1)− p ( Θ ( p−1 )( α β) β β p+1 dx ∧ dα + αp−1 β 2 dy ∧ dβ) p p+1 1 1 α p−1 Θp−1 p−2 p−1 + 1)− p ( (( α dx ∧ dα (p−1)2 ( β ) β) β p+1 αp−2 (p − 1)α p−1 Ω p−2 + αp−1 β p (p − 1)β dy ∧ dβ). (2.22)

Indeed,

Θp−1 Ωp−1 = β p+1 αp−2 αp−1 β p

(2.23)

Θp−1 β = Ωp−1 α

(2.24)

since

by (2.21).

ON EXPLICIT HOLMES-THOMPSON AREA FORMULA IN INTEGRAL GEOMETRY

||(Θ, Ω)||p = 1,

Therefore, using (2.23), (2.24) and

dr ∧ dΘ

p p−1

1 p−1

1 α (p−1)2 ( β )

− p+1 p

we have

p−1

=

Θ (( α + 1) β) β p+1 αp−2 p−2 ((p − 1)α dx ∧ dα + (p − 1)β p−2 dy ∧ dβ) p+1 p 1 α p−1 Θp−1 1 p−1 + 1)− p (( α (p−1)2 ( β ) β) β p+1 αp−2 ω0

=

1 (p−1)2

=

9

1

p−1 Θp−1 (α β) p

p−1 +1) (( α β)

=

1 (p−1)2

=

1 (p−1)2 (

= = =

p+1 p

p+1 Ω p (( Θ ) +1) p

ω0

β p+1 αp−2

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

ΩΘ )p+1 (

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

ω0

(2.25)

ω0

ω

ω

ω ,

Thus we have shown

(p−1)(2p−1)

||(Θ, Ω)||p(p−1)

dr ∧ dΘ =

(p − 1)2 Θp(p−2) Ωp2 −3p+1

ω0 ,

(2.26)



which implies (2.19) in the claim. So from (2.18) and (2.19) we see the symplectic structure on mined by the Minkowski metric

|| · ||p

on

R

2

Gr1 (R2 )

is deter-

.

3. Integral Geometry on Length in Minkowski Space The length of a straight line segment in the canonical contact form be the vector from

x

to

y,

c(t) := (x +

α

dF ( Therefore,

´

c where

α

L(xy)

=

→ ´ F (− xy) 0

can be obtained by integrating

introduced in Section 1.2. For any

x, y ∈ R2 ,

let

− → xy

and

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

be a straight line segment in the useful fact that

(R2 , F )

T ∗ R2 .

(3.1)

By the positive homogeneity of F, one can get

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

− →

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

is the length of

→ = F (− xy)

(3.2)

= L(xy),

(3.3)

xy .

Here let us introduce a general denition in integral geometry rst.

Denition 3.1.

A Crofton measure

φ

for a degree

k

measure

Φ

on

(Rn , F )

is n a measure on Grn−k (R ) (Denition 1.3), such that it satises the Crofton-type

ˆ

formula

Φ(M ) = P ∈Grn−k (Rn ) n for any compact convex subset M (R , F ).

#(M ∩ P )Φ(P )

(3.4)

10

LOUIS Y. LIU

Figure 3.1. From

SR2

to

Gr1 (R2 )

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 [4].

Proof.

ψ

symplectic form

 Let

'

Gr1 (R2 ) → T SF T SF embeddedin T ∗ R2 .

From Section 2, we know

ω0

as

◦

l ∈ Gr1 (R2 ) : l ∩ xy 6= φ

S :=

which is a cylinder, and it has a

,

Cx

and

Cy

be the family of oriented lines

+ − , lxy that are the two Cx ∩ Cy = lxy oriented lines connecting x and y , and ∂S = Cx ∪ Cy .  2 2 Let R := ξ ∈ SR ⊂ T R : 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 , 2 where p is the natural projection from SR to Gr1 (R2 ).   0+ − → 0 0 2 = F (xy Additionally, let Cx = ξx : ξx ∈ Sx R , Cy = ξy : ξy ∈ Sy R2 , lxy − → ∈ xy) passing through

0

x

and

y

respectively, then

− →

0

0

0

+ 2 − = − F (xy Sx R2 , and lxy − → ∈ Sy R , then p maps Cx , Cy , lxy xy) − and lxy respectively, see Figure 3.1 on page 10.

´

0

−

and lxy to

+ Cx , Cy , lxy

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

α = 0 because of the xed base points for any C0 with (3.3), we obtain ´

S

´

|ω0 | =

p(R)

Therefore, for any rectiable curve

L(γ) =

1 4

ˆ

γ

0

C 0 ⊂ Cx

´ |ω0 | = R |p∗ ω0 | ´ = R´ |ω| = 2 l0 + ∪l0 − α xy xy = 4L(xy). in

(R2 , F ),

the length of

#(γ ∩ l)|ω0 |,

.

3.3

(3.5)

γ, (3.6)

l∈Gr1 (R2 )



which is the desired claim.

Remark

0

or Cy , and combining

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

ON EXPLICIT HOLMES-THOMPSON AREA FORMULA IN INTEGRAL GEOMETRY

metric

F,

P ⊂ Rn containing xy for any x, y ∈ Rn , ˆ 1 L(xy) = #(xy ∩ l)|ω0 |. 4 l∈Gr1 (P )

we choose a plane

11

then (3.7)

4. Volume of Hypersurfaces A standard denition of Holmes-Thompson volume in Minkowski space

(Rn , F )

is given and its importance in Finsler geometry and integral geometry is illustrated in [3]. The Holmes-Thompson volumes are dened as follows.

Denition 4.1.

N

Let

be a

k -dimensional

manifold and

D∗ N := {ξx ∈ T ∗ N : F ∗ (ξx ) 6 1} , where

F

∗

is the dual norm in (1.7), be the codisc bundle of

Holmes-Thompson volume is dened as

1 k

volk (N ) := where

k

(4.1)

is the Euclidean volume of

N

k -th

, then the

ˆ |ω k |,

(4.2)

D∗ N

k -dimensional

canonical symplectic form on the cotangent bundle of

Euclidean ball and

ω

is the

N.

ˆ 0 are the natural symplectic forms on Λ ∈ Grk (Rn ) for some k ≤ n, ω0 and ω Gr1 (Λ) constructed in the way described in Section 2. The relation between ω0 and ω ˆ 0 is shown in the following Let

Gr1 (Rn ) and

Lemma 4.2. Proof.

i∗ ω0 = ω for i : Gr1 (Λ) ,→ Gr1 (Rn ).

First consider the diagram ϕF ∗

'

ϕF

ˆi

'

S ∗ Λ → SΛ ,→ SRn → S ∗ Rn .

(4.3)

∗

α ˆ ξ (X) := ξ(ˆ π0∗ X) for X ∈ Tξ S Λ on S Λ in π ˆ0 : S ∗ Λ → Λ is the natural projection, and dene ω ˆ := dˆ α

We have a canonical contact form diagram (4.3), where on

∗

S ∗ Λ. Let j = ϕF ◦ ˆ i ◦ 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) in which

α

and

ω

on

S ∗ Rn

(4.4)

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 dierentiating (4.5). Next, let direction

ξ¯x ,

pˆ

be the projection taking

and similarly for

S∗Λ

ϕF ∗

'

p

ξ¯x ∈ SΛ

to the line passing

x

with the

which is described in (2.1). Consider the diagram

SΛ ↓ pˆ Gr1 (Λ)

ˆi

,→

SRn ↓p

ϕF

'

S ∗ Rn (4.7)

i

,→ Gr1 (Rn )

obtained by combining diagram (2.1) and (4.4). By the denitions of the maps in (4.7), we know the diagram is commutative. By Theorem 2.3, we have

p∗ ω0 = ϕ∗F ω

12

LOUIS Y. LIU

pˆ∗ ω ˆ 0 = ϕ∗F ω ˆ.

and

Combining with (4.6) and the commutativity of the diagram

i∗ ω0 = ω ˆ0.

(4.7), we obtain the desired claim Suppose

N

is a hypersurface in

Proposition 4.3.

voln−1 (N ) =

symplectic form on Gr1 (Rn ).

(Rn , F ), ´ 1



then we have the following

l∈Gr1 (Rn )

2n−1

#(N ∩ l)|ω0n−1 |, where ω0 is the

This idea of intrinsic proof is given by Dr. Joseph H. G. Fu.

Proof.

It suces to prove the claim in the case when

generality, assume

N ⊂ Rn−1 ⊂ Rn

N

is ane. Without loss of

is compact and convex with smooth boundary.

Consider the following diagram ϕF ∗

ˆi

∼ =

ϕF

∼ =

i

π

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

i

and

k

are embeddings, and

π := p ◦ ϕ−1 F = p ◦ ϕF ∗

(4.8)

is a projection from

diagram (2.1). As

N

is a

is dened as

(n − 1)-dimensional α ˆ ξ (X) := ξ(ˆ π0∗ X)

α ˆ on S ∗ N π ˆ0 : S N → N is the

manifold, the canonical contact form for

∗

X ∈ Tξ S N ,

where

∗

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). X ∈ Tξ S ∗ N , which implies (ˆi ◦ j)∗ α = ˆi∗ j ∗ α = α ˆ, ω ˆ := dˆ α and ω is introduced in Section 1.2.

for any where

Applying Stokes' theorem, we have

´

D∗ N

ω ˆ n−1

=

´

∂(D ∗ N )

and then

(4.9)

(i ◦ j)∗ ω = ω ˆ

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

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

=

(4.11) where

 ∗ n S+ R = ξ ∈ S ∗ Rn : ξ(v0 ) > 0, v0

satises dF (v0 )(v)

=0

for all v

∈ SRn−1 . (4.12)

Therefore,

ˆ

ˆ ω ˆ n−1 =

D∗ N

ω n−1 .

(4.13)

∗ Rn ∩π −1 (N ) S+ 2

Now let us consider the upper half space of geodesics in

(Rn , F ),

 Gr1+ (Rn ) := l(x, η) : dF (η)(η0 ) > 0, η0 satises dF (η0 )(v) = 0

for all v

∈ SRn−1 . (4.14)

ON EXPLICIT HOLMES-THOMPSON AREA FORMULA IN INTEGRAL GEOMETRY

Since

π ∗ ω0 = ω , ´

then we get

∗ Rn ∩π −1 (N ) S+ 0

ω n−1

= = =

´ ∗

´S+ R ´π

= = =

π ∗ ω0n−1

−1 (l)∈S ∗ Rn ∩π −1 (N ) 0 +

l∈Gr1+ (Rn )

Combining with (4.13), we obtain

voln−1 (N )

n ∩π −1 (N ) 0

#(N ∩ l)ω0n−1

(4.15)

#(N ∩ l)ω0n−1 .

´

1 ˆ n−1 n−1 ´D ∗ N ω n−1 1 n−1 l∈Gr + (Rn ) #(N ∩ l)ω0 1 ´ n−1 1 |, 2n−1 l∈Gr1 (Rn ) #(N ∩ l)|ω0

(4.16)



that nishes the proof. 5.

k -th

13

Holmes-Thompson Volume and Crofton Measures

Let us introduce a general fact rst. Busemann constructed all projective metrics

F

for projective Finsler space

(Rn , F ),

and it was also proved in [12] 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 form on S n−1 and f is some continuous function on S n−1 × R. ξ∈S n−1

In fact, for the case that

(Rn , F )

tivity of cosine transform,

is Minkowski, we can use a theorem on surjec-

ˆ

C(f )(·) =

|hξ, ·i|f (ξ)Ω0 ,

(5.1)

ξ∈S n−1 of even functions from Chapter 3 of [9],

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 v=− xy ˆ 1 F (x, v) = #(xy ∩ l)|ω0 |. ωn−1 l∈Gr1 (R2 )

On the other hand, for any

In fact, there is a relation between

Ω0

and

ω0 .

3.2 we know (5.3)

Considering the following double

bration

π

where

π

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

Proposition 5.3. the double braton origin.

(5.4) have

GT (f Ω0 ∧ dr) = ω0 , where GT is the Gelfand transform for and r is the Euclidean distance of a hyperplane H to the

(5.4)

14

LOUIS Y. LIU

Proof.

2-plane Π ⊂ Rn , we know that the length of xy , ˆ 1 L(xy) = #(xy ∩ l)|ω0 |Gr1 (Π) . (5.5) 4 l∈Gr1 (Π) n o where Gr1 (Π) := l ∈ Gr1 (Rn ) : l ⊂ Π . n o −1 Let I = l ∈ Gr1 (Π) ⊂ Gr1 (Rn ) : xy ∩ l 6= φ and GH := π1 (π2 (H)) for H ∈ For any

Grn−1 (Rn ).

x, y

in any

By the fundamental theorem of Gelfand transform, Theorem 1.5,

ˆ

ˆ

#(I ∩ GH )|f Ω0 ∧ dr| = H∈Grn−1 (Rn ) Therefore,

´

l∈Gr1 (Π)

|GT (f Ω0 ∧ dr)|.

(5.6)

I

#(xy ∩ l)|GT (f Ω0 ∧ dr)| = = =

´ |GT (f Ω0 ∧ dr)| ´I #(I ∩ GH )|f Ω0 ∧ dr| ) ´H∈Grn−1 (Rn− → (ξ)Ω |hξ, xyi|f 0 n−1 ξ∈S (5.7)

since

(Rn )

Grn−1 ˆ

∼ = S n−1 × R.

By (5.2)and (5.5) we thus obtain

ˆ

#(xy ∩ l)|GT (f Ω0 ∧ dr)| =

#(xy ∩ l)|ω0 |,

l∈Gr1 (Π) which implies

(5.8)

l∈Gr1 (Π)

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 [9] Groemer shows by using condensed harmonic expansion and Parseval's equation, that

C(f1 ) = C(f2 ) i f1+ =

+ 1 (−v) f1+ (v) = f1 (v)+f and similarly for f2 , for any bounded integrable 2 n−1 functions f1 and f2 on S .) Now dene a basis for Tl Gr1 (Rn ), the tangent space of Gr1 (Rn ) at l ∈ Gr1 (Rn ).

f2+ ,

where

ψ

Gr1 (Rn ) ' T SFn−1 from Section 2. Let {ei : i = 1, · · · , n} be the basis n for R , and curve γ i with γ i (t) = l+tei for i = 1, · · · , n−1, where l ∈ Gr1 (Rn ), and 0 then dene ei := γ i (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 dene curves γ i (t) = ψ −1 (vi (t)) for  0 i = 1, · · · , n − 1, thus we can dene ei := γ i (0). Then ei , ej : i, j = 1, · · · , n − 1 is a basis for Tl Gr1 (Rn ). Note that

Here we have four cases to discuss. First of all, one can obtain the fact

GT (f Ω0 ∧ dr)(ei , ei ) = ω0 (ei , ei ) by choosing a plane

Πi

with the tangent space of

Gr1 (Πi )

(5.9) spanned by

ei

and

ei

for

i = 1, · · · , n − 1. On the other hand, in the double bration (5.4),

π2 |π−1 (Lij ) , in which Lij 1

is be the

Gr1 (Rn ) obtained by translation along ei or ej for i, j = 1, · · · , n − 1, is not −1 a submersion from π1 (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 ) = lines in

ON EXPLICIT HOLMES-THOMPSON AREA FORMULA IN INTEGRAL GEOMETRY

´ π1−1 (l) thus

π2∗ (f Ω0 ∧ dr)(ei , ej ) = 0

for

i, j = 1, · · · , n − 1,

and obviously

15

ω0 (ei , ej ) = 0,

GT (f Ω0 ∧ dr)(ei , ej ) = ω0 (ei , ej ) = 0 (5.10) 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 −1 ¯ n (5.4) is not a submersion from π1 (L ij ) to Grn−1 (R ) either, and it also can be ∗ 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 for

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

(5.11)

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

(5.12)

Gr1 (Rn ).

on



One can use the diagonal intersection map and Gelfand transform by following [6] to construct Crofton measure for the Let

Ωn−1 := f Ω0 ∧ dr

k -th

Holmes-Thompson volume.

and dene a map

k

π : Grn−1 (Rn ) \4k → Grn−k (Rn ) π((H1 , · · · , Hk )) = H1 ∩ · · · ∩ Hk , 4k = {(H1 , · · · , Hk ) : dim(H1 ∩ · · · ∩ Hk ) > n − k} π∗ Ωkn−1 . where

(5.13) and then let

Ωn−k :=

Now consider the following double bration,

π1,k

where

π2,k

Gr1 (Rn ) ← Ik → Grn−k (Rn ), o n Ik = (l, S) ∈ Gr1 (Rn ) × Grn−k (Rn ) : l ⊂ S . Then

(5.14) we have the following

proposition about the Gelfand transform on (5.14)

Proposition 5.4. Proof.

GT (Ωn−k ) = ω0k for 1 ≤ k ≤ n − 1.

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

← π ˜1-

Ik ↑π ˜ H

in which

π ˜ : H → Ik

is dened by

π2,k

→

π ˜2

→

Grn−k (Rn ) ↑π Grn−1

(5.16)

k (Rn ) ,

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)

16

LOUIS Y. LIU

By manipulating the map

π ˜ 2 = π2 × · · · × π2 , | {z }

the product of

k

copies of the map

k

π ˜1 × π ˜2 : H ∗ k k Gr1 × π ˜1∗ π ˜2 Ωn−1 = ω0 . ∗ Thus, by the commutativity of the diagram (5.16) we obtain π1,k∗ π2,k Ωn−k k ω0 . π2 ,

applying Fubini theorem for (5.17) and using the fact that

→

k Gr1 (Rn ) is an immersion, one can infer

(Rn )

k -th

In order to study the

k + 1-dimensional

= 

Holmes-Thompson volume, one can restrict on some

at subspace. So x

S ∈ Grk+1 (Rn )

and then dene a map by

intersection

πS : Grn−k (Rn ) \ 4(S) → Gr1 (S) πS (H n−k ) = H n−k ∩ S for

(5.19)

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. Proof.

(πS )∗ Ωn−k = ω0k |Gr1 (S) , for 1 ≤ k ≤ n − 1.

From Proposition 5.4, we know that

bration

π2,k

∗ π1,k∗ π2,k Ωn−k = ω0k

for the double

π1,k

Grn−k (Rn ) ← Ik → Gr1 (Rn ).

Therefore, one can obtain by the denition 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 ´ Then volk (N ) =

Proof.

1 2k

N is a k -dimensional submanifold in (Rn , F ). #(N ∩ P )|Ωn−k | for 1 ≤ k ≤ n − 1. P ∈Grn−k (Rn )

By Proposition 4.3, the claim is true for hypersurface case.

It is sucient to show the claim for the case when

Grk+1 (Rn ).

N ⊂ S

for some

S ∈

We obtain by Proposition 4.3 and Proposition 5.5,

volk (N )

= = =

1 2k 1 2k 1 2k

´ #(N ∩ l)|ω0k | ´l∈Gr1 (S) #(N ∩ l)|(πS )∗ Ωn−k | ´l∈Gr1 (S) #(N ∩ P )|Ωn−k |. P ∈Grn−k (Rn )

(5.22)



as desired. 6. Length and Related The classic Crofton formula is

1 Length(γ) = 4

ˆ

∞

ˆ

2π

#(γ ∩ l(r, θ))dθdr 0

θ is the angle of the r is its distance to the origin. Let 2 in R by Gr1 (R2 ) .

for any rectiable curve in Euclidean plane, where the oriented line the ane

l

to the

x-axis

1-Grassmannians

and

(lines)

(6.1)

0 normal of us denote

As for Minkowski plane, it is a normed two dimensional space with a norm

F (·) = || · ||

, in which the unit disk is convex and

F

has some smoothness.

ON EXPLICIT HOLMES-THOMPSON AREA FORMULA IN INTEGRAL GEOMETRY

17

Two of the key tools used to obtain the Crofton formula for Minkowski plane are the cosine transform and Gelfand transform. Let us explain them one by one rst and see their connection next. A fact from spherical harmonics about cosine

S1

transform is there is some even function on

F (·) = if

F

is an even

C4

function on

S1.

1 4

ˆ

such that

|hξ, ·i|g(ξ)dξ,

(6.2)

S1

A good reference for this is [9]. As for Gelfand

transform, it is the transform of dierential forms and densities on double brations,

n o π π R2 ←1 I →2 Gr1 (R2 ), where I := (x, l) ∈ R2 × Gr1 (R2 ) : x ∈ l is incidence relations and π1 and π2 are projections. A formula one can take as

for instance, the

an example of the fundamental theorem of Gelfand transform is the following

ˆ

ˆ π1∗ π2∗ |Ω|

Ω := g(θ)dθ ∧ dr.

where

Proof.

π1∗ (v 0 ) = v ,

such that

Ω = dθ ∧ dr.

´

γ mula.

π1∗ π2∗ |Ω| = 4Length(γ) =

When

Ω = f (θ)dθ ∧ dr,

For any

v ∈ Tx γ ,

since there is some

then

(π1∗ π2∗ |Ω|)x (v)

So

(6.3)

l∈Gr1 (R2 )

But we give a direct proof here.

First, consider the case of

v 0 ∈ Tx0 I ,

#(γ ∩ l)|Ω|,

=

γ

= = = = = =

´

´ ( −1 π ∗ |Ω|) (v) ´ π1 (x) 2 ∗ x (π |Ω|)x0 (v 0 ) −1 ´x0 ∈π1∗ (x) 2 0 ) ´S 1 (π2 |dθ ∧ dr|)(v 0 |dr(π (v ))|dθ 2∗ 1 ´S |hv, θi|dθ S1 4|v|.

l∈Gr1 (R2 )

#(γ ∩ l)|Ω|

we just need to replace

dθ

(6.4)

by the classic Crofton for-

by

g(θ)dθ

in the equalities in



the rst case. Moreover, from the above proof and (6.2), for any curve

γ(t) : [a, b] → R2

dierentiable almost everywhere in the Minkowski space,

ˆ

ˆ π1∗ π2∗ |Ω| =

γ

ˆ

b

(π1∗ π2∗ |Ω|)(γ 0 (t))dt =

b

4F (γ 0 (t))dt = 4Length(γ),

a

(6.5)

a

so then by (6.3) we know

Length(γ) =

1 4

ˆ #(γ ∩ l)|g(θ)dθ ∧ dr|

(6.6)

l∈Gr1 (R2 )

for Minkowski plane.

2 HT ´ (U ) of2 a measurable set U in a Minkowski dened as HT (U ) := |ω0 | , where natural symplectic D∗ U  ω0 is the 2 ∗ ∗ 2 ∗ the cotangent bundle of R and D U := (x, ξ) ∈ T R : F (ξ) ≤ 1 . To

The Holmes-Thompson Area plane is form on

2

1 π

study it from the perspective of integral geometry, we need to introduce a symplectic form

ω

on the space of ane lines

Gr1 (R2 ),

that one can see [1].

18

LOUIS Y. LIU

7. HT Area and Related

´Now let's see the Crofton formula for Minkowski plane, which is Length(γ) = 1 4 Gr1 (R2 ) #(γ ∩ l)|ω|. To prove this, it is sucient to show that it holds for for any straight line segment L : [0, ||p2 − p2 ||] → R2 , L(t) = p1 + starting at

p1

and ending at

p2

in

R2 .

p2 − p1 t, ||p2 − p1 ||

(7.1)

First, using the dieomorphism between the

circle bundle and co-circle bundle, which is

ϕF : SR2 → S ∗ R2 ϕF (x, ξ) = (x, dFξ ), we can obtain a fact that

´

L×

n

p2 −p1 ||p2 −p1 ||

o

ϕ∗F α0

(7.2)

´

=

n o α p −p ϕF (L× ||p2 −p1 || ) 0 2 1 ´ ||p2 −p1 || α0dF p2 −p1 0

=

´ ||p2 −p1 ||

=

0

||p2 −p1 ||

dF

p2 −p1 ||p2 −p1 ||

−p1 (( ||pp22 −p , 0))dt 1 ||

(7.3)

1 ( ||pp22 −p −p1 || )dt,

where α0 is the tautological one-form, precisely α0ξ (X) := ξ(π0∗ X) for any X ∈ Tξ T ∗ R2 , and dα0 = ω0 . Applying the the basic equality that dFξ (ξ) = 1, which is 2 derived from the positive homogeneity of F , for all ξ ∈ SR , the above quantity ´ ||p2 −p1 || becomes 1dt, which equals to ||p2 − n p1 ||. 0 o  0 ∗ 2 Let R := ξx ∈ S R : x ∈ p1 p2 and T = l ∈ Gr1 (R2 ) : l ∩ p1 p2 6= Ø , and p

S ∗ R2 to Gr1 (R2 ). p0∗ ω = ω0 , ´ ´ ´ ´ |ω| = p0 (R) |ω| = R |p0∗ ω| = R |ω0 | T ´ ´ = | ´R+ ω0 | + | R ´ − ω0 | = | ∂R+ α0 | + | ∂R− α0 | = 4||p2 − p1 ||.

is the projection (composition) from Apply the above fact and

(7.4)

Thus we have shown the Crofton formula for Minkowski plane. Furthermore, combining with (6.6), we have

1 4 where

ˆ

#(γ ∩ l)|Ω| = l∈Gr1 (R2 )

Ω = g(θ)dθ ∧dr.

1 4

ˆ

#(γ ∩ l)|ω|,

(7.5)

Gr1 (R2 )

Then, by the injectivity of cosine transform in [9],

|Ω| = |ω|.

To obtain the HT area, one can dene a map

˜ → R2 π : Gr1 (R2 ) × Gr1 (R2 ) \ 4 0 0 π(l, l ) = l ∩ l , where

˜ := {(l, l0 ) : l k l0 4

? Theorem 7.1.

= l0 },

or l

(7.6)

extended from Alvarez's construction of taking

intersections, [ ]. The following theorem can be obtained.

HT 2 (U ) =

1 2π

subset U of a Minkowski plane. Proof.

On one hand,

1 π

ˆ D∗ U

ω02

´

x∈R2

1 = π

χ(x ∩ U )|π∗ Ω2 | for any bounded measurable

ˆ ∂D ∗ U

ω02

1 = π

ˆ α0 ∧ ω0 . S∗ U

(7.7)

ON EXPLICIT HOLMES-THOMPSON AREA FORMULA IN INTEGRAL GEOMETRY

n o TU := ((l, l0 ) ∈ Gr1 (R2 ) × Gr1 (R2 ) : l ∩ l0 ∈ U , ˆ ˆ ˆ 1 1 1 2 2 χ(x ∩ U )π∗ Ω = π∗ ω = ω2 . π x∈R2 π U π TU

19

On the other hand, let

Let

T∗ U := {(ξx , ξx0 ) : ξx , ξx0 ∈ Sx∗ U }, 0

0 −1

(p × p ) Therefore

1 π

´ TU

ω2

then

(TU ) = T∗ U \ {(ξx , ξx ) : ξx ∈ Sx∗ U } . = = = =

1 π 1 π 2 π 2 π

(7.8)

´ p0∗ ω 2 ´T∗ U \{(ξx ,ξx ):ξx ∈Sx∗ U } 2 ω ´T∗ U \{(ξx ,ξx ):ξx ∈Sx∗ U } 0 α ∧ ω0 ´{(ξx ,ξx ):ξx ∈Sx∗ U } 0 α ∧ ω . 0 S∗ U 0

(7.9)

(7.10)

So the claim follows from (7.7),(7.8) and (7.10).

Acknowledgement.



Thanks to J. Fu for some helpful discussions on this subject.

References

[1] J.C. Álvarez Paiva, Symplectic Geometry and Hilbert's Fourth Problem, J. Dierential Geom. Volume 69, Number 2 (2005), 353-378. [2] Semyon Alesker, Theory of Valuations on Manifolds: A Survey, Geometric And Functional Analysis, Volume 17, Number 4, November, 2007. [3] J. C. Álvarez Paiva and Gautier Berck, What is wrong with the Hausdor measure in Finsler spaces, Advances in Mathematics Volume 204, Issue 2, 20 August 2006, Pages 647-663. [4] J. C. Álvarez Paiva and C.E. Durán, An introduction to Finsler geometry, Publicaciones de la Escuela Venezolana de Matématicas, Caracas, Venezuela, 1998. [5] J. C. Álvarez Paiva and E. Fernande, Crofton formulas and Gelfand transforms. Accept
e pour publication dans Selecta Mathematica, 21pp. [6] J. C. Álvarez Paiva and E. Fernande, Crofton formulas for projective Finsler spaces, Electronic Research Announcements of the AMS, 4 (1998) 91-100. [7] Andreas Bernig, Valuations with Crofton formula and Finsler geometry, Advances in Mathematics, 2007; 210 (2). [8] Shiing-Shen Chern, Zhongmin Shen, Riemann-Finsler Geometry, World Scientic, Published 2005. [9] H. Groemer, Geometric Applications of Fourier Series and Spherical Harmonics, Cambridge University Press, 1996. [10] Dan Klain, Even valuations on convex bodies, Transactions of the American Mathematical Society, 352 (2000), no. 1, 71-93. [11] Rolf Schneider, Convex Bodies: The Brunn-Minkowski Theory, Cambridge University Press, 1993 [12] Rolf Schneider, Zur einem Problem von Shephard über die Projektionen konvexer Körper (in German), Math. Z. 1967, 101: 7182. Department of Mathematics, University of Georgia, Athens, GA 30602, U.S.A. [email protected]