6. Application to generalized length and related

For any rectifiable curve γ in the Euclidean plane, the classic Crofton formula is

$$\text{Length}(\gamma) = \frac{1}{4} \int\_0^\infty \int\_0^{2\pi} \#(\gamma \text{cl}(r, \theta)) \text{d}\theta \text{d}r,\tag{27}$$

where θ is the angle from the x-axis to the normal of the oriented line l and r is the distance form the origin to l. Let us denote the affine l-Grassmannians consisting of lines in R<sup>2</sup> by Gr1ðR<sup>2</sup>Þ.

As for Minkowski plane, it is a normed two dimensional space with a norm Fð�Þ ¼ jj � jj, in which the unit disk is convex and F has some smoothness.

Two significant and useful tools that are used to obtain the Crofton formula for Minkowski plane are the cosine transform and Gelfand transform. Let us explain them one by one first and see the connections between them later. A important fact or result from spherical harmonics about cosine transform is that there is some even function on S<sup>1</sup> such that

$$F(\cdot) = \int\_{\mathcal{S}^1} |\langle \xi, \cdot \rangle| \mathcal{g}(\theta) \mathbf{d}\theta,\tag{28}$$

if F is an even C<sup>4</sup> function on S<sup>1</sup> . A great reference for this would be [24] by Groemer. As for Gelfand transform, it is the transform of differential forms and densities on double fibrations, for instance, R<sup>2</sup>← π1 I! π2 Gr1ðR<sup>2</sup>Þ, where <sup>I</sup> :<sup>¼</sup> n <sup>ð</sup>x;lÞ<sup>∈</sup> <sup>R</sup><sup>2</sup> · Gr1ðR<sup>2</sup> Þ : x ∈l o is the incidence relations and π<sup>1</sup> and π<sup>2</sup> are projections. A formula one can take as an example of the fundamental theorem of Gelfand transform is the following:

$$\int\_{\mathcal{V}} \pi\_{1\*} \pi\_2^\* |\mathcal{Q}| = \int\_{l \in \overline{Gr\_1(\mathbb{R}^2)}} \#(\mathcal{\gamma} \mathbb{T}l) |\mathcal{Q}|, \tag{29}$$

where Ω :¼ gðθÞdθ∧dr. However, here we provide a direct proof for this fundamental theorem of Gelfand transform.

Proof. First, consider the case of <sup>Ω</sup> <sup>¼</sup> <sup>d</sup>θ∧dr. For any <sup>v</sup><sup>∈</sup> Txγ, since there is some <sup>v</sup>′ <sup>∈</sup>Tx′I, such that <sup>π</sup>1�ðv′ Þ ¼ v, then

$$\begin{aligned} (\pi\_{1\*}\pi\_2^\*|\varOmega|)\_x(v) &= \quad (\int\_{\pi\_1^\*(x)} \pi\_2^\*|\varOmega|)\_x(v) \\ &= \quad \int\_{x' \in \pi\_1^\*(x)} (\pi\_2^\*|\varOmega|)\_{x'}(v') \\ &= \quad \int\_{S^1} (\pi\_2^\*|\varOmega \theta \land \varGamma|)(v') \\ &= \quad \int\_{S^1} |\varGamma(\pi\_{2\*}(v'))|\varOmega \\ &= \quad \int\_{S^1} |\langle v, \theta \rangle| \mathrm{d}\theta \\ &= \quad \nwarrow 4|v|. \end{aligned} \tag{30}$$

Thus, we have

6. Application to generalized length and related

which the unit disk is convex and F has some smoothness.

if F is an even C<sup>4</sup> function on S<sup>1</sup>

π1 I! π2

theorem of Gelfand transform is the following:

for instance, R<sup>2</sup>←

of Gelfand transform.

Þ ¼ v, then

that <sup>π</sup>1�ðv′

Gr1ðR<sup>2</sup>Þ.

36 Lagrangian Mechanics

For any rectifiable curve γ in the Euclidean plane, the classic Crofton formula is

4 ð∞ 0

ð<sup>2</sup><sup>π</sup> 0

where θ is the angle from the x-axis to the normal of the oriented line l and r is the distance form the origin to l. Let us denote the affine l-Grassmannians consisting of lines in R<sup>2</sup> by

As for Minkowski plane, it is a normed two dimensional space with a norm Fð�Þ ¼ jj � jj, in

Two significant and useful tools that are used to obtain the Crofton formula for Minkowski plane are the cosine transform and Gelfand transform. Let us explain them one by one first and see the connections between them later. A important fact or result from spherical harmonics

Gelfand transform, it is the transform of differential forms and densities on double fibrations,

tions and π<sup>1</sup> and π<sup>2</sup> are projections. A formula one can take as an example of the fundamental

where Ω :¼ gðθÞdθ∧dr. However, here we provide a direct proof for this fundamental theorem

Proof. First, consider the case of <sup>Ω</sup> <sup>¼</sup> <sup>d</sup>θ∧dr. For any <sup>v</sup><sup>∈</sup> Txγ, since there is some <sup>v</sup>′ <sup>∈</sup>Tx′I, such

ð π−<sup>1</sup> <sup>1</sup> ðxÞ π� <sup>2</sup>jΩjÞxðvÞ

x′ ∈ π−<sup>1</sup> <sup>1</sup> ðxÞ ðπ�

<sup>l</sup> <sup>∈</sup>Gr1ðR<sup>2</sup><sup>Þ</sup>

<sup>ð</sup>x;lÞ<sup>∈</sup> <sup>R</sup><sup>2</sup> · Gr1ðR<sup>2</sup>

n

ð

#ðγ∩lðr;θÞÞdθdr; (27)

j〈ξ; � 〉jgðθÞdθ; (28)

. A great reference for this would be [24] by Groemer. As for

2jΩjÞx′ðv′

ÞÞjdθ

Þ

2jdθ∧drjÞðv′

<sup>j</sup>drðπ2�ðv′

j〈v;θ〉jdθ

Þ

Þ : x ∈l o

#ðγ∩lÞjΩj, (29)

is the incidence rela-

(30)

LengthðγÞ ¼ <sup>1</sup>

about cosine transform is that there is some even function on S<sup>1</sup> such that

Gr1ðR<sup>2</sup>Þ, where <sup>I</sup> :<sup>¼</sup>

ð γ π<sup>1</sup>�π�

ðπ<sup>1</sup>�π�

Fð�Þ ¼ ð S1

<sup>2</sup>jΩj ¼

<sup>2</sup>jΩjÞxðvÞ¼ð

¼ ð

¼ ð S1 ðπ�

¼ ð S1

¼ ð S1

¼ 4jvj:

$$\int\_{\mathcal{V}} \pi\_{1\*} \pi\_2^\* |\Omega| = 4 \text{Length}(\mathcal{V}) = \int\_{l \in Gr\_1(\mathbb{R}^2)} \# (\gamma \cap l) |\Omega| \tag{31}$$

by using the classic Crofton formula.

For the general case of Ω ¼ fðθÞdθ∧dr, we just need to substitute dθ by gðθÞdθ in the equalities in the first case.

Furthermore, we can also see, from the above proof and eq:exist, that

$$\int\_{\mathcal{V}} \pi\_{1\*} \pi\_2^\* |\mathcal{Q}| = \int\_a^b (\pi\_{1\*} \pi\_2^\* |\mathcal{Q}|)(\dot{\boldsymbol{\chi}}^\prime(t)) \mathbf{d}t = \int\_a^b 4F(\dot{\boldsymbol{\chi}}^\prime(t)) \mathbf{d}t = 4\text{Length}(\boldsymbol{\chi}),\tag{32}$$

for any curve <sup>γ</sup>ðt<sup>Þ</sup> : <sup>½</sup>a;b� ! <sup>R</sup><sup>2</sup> differentiable almost everywhere in the Minkowski space. Therefore, by using (29), we obtain that

$$\text{Length}(\gamma) = \frac{1}{4} \int\_{l \in \text{Gr}\_1(\mathbb{R}^2)} \#(\gamma \cap l) |g(\theta) \,\text{d}\theta \wedge \text{d}r| \tag{33}$$

for Minkowski plane.

The Holmes-Thompson area HT<sup>2</sup> ðUÞ of a measurable set U in a Minkowski plane is defined as HT<sup>2</sup> ðUÞ :¼ <sup>1</sup> π ð D�U jω0j 2 , where ω<sup>0</sup> is the natural symplectic form on the cotangent bundle of R<sup>2</sup> and D� U :¼ fðx;ξÞ∈T� R<sup>2</sup> : F� ðξÞ ≤ 1g. To study it from the perspective of integral geometry, we need to introduce a symplectic form <sup>ω</sup> to the space of affine lines Gr1ðR<sup>2</sup><sup>Þ</sup> and construct an invariant measure based on ω.

#### 7. Application to HT area and related

Now let us see the Crofton formula for Minkowski plane, which is

$$\text{Length}(\gamma) = \frac{1}{4} \left| \frac{}{Gr\_1(\mathbb{R}^2)} \# (\gamma \cap l) | \alpha| \,. \tag{34}$$

To prove this, it is sufficient to show that it holds for any straight line segment

$$L: [0, ||p\_2 - p\_2||] \to \mathbb{R}^2, L(t) = p\_1 + \frac{p\_2 - p\_1}{||p\_2 - p\_1||}t,\tag{35}$$

starting at <sup>p</sup><sup>1</sup> and ending at <sup>p</sup><sup>2</sup> in <sup>R</sup>2. First, using the diffeomorphism between the circle bundle and co-circle bundle, which is

$$\begin{cases} \varphi\_F: \mathbb{S}\mathbb{R}^2 \to \mathbb{S}^\*\mathbb{R}^2\\ \varphi\_F(\mathbf{x}, \xi) = (\mathbf{x}, \mathbf{d}F\_{\xi}), \end{cases} \tag{36}$$

we can obtain a fact that

$$\begin{array}{ll}\int\_{L\times\left\{\frac{p\_{2}\cdot p\_{1}}{\|p\_{2}\cdot p\_{1}\|}\right\}}\phi\_{F}^{\*}\alpha\_{0} &=\int\_{\varphi\_{I}(L\times\left\{\frac{p\_{2}\cdot p\_{1}}{\|p\_{2}\cdot p\_{1}\|}\right\})}\alpha\_{0} \\ &=\int\_{0}^{||p\_{2}-p\_{1}||}\alpha\_{0 \text{d}F}\,\_{\frac{p\_{2}\cdot p\_{1}}{\|p\_{2}\cdot p\_{1}\|}}((\frac{p\_{2}-p\_{1}}{\|p\_{2}-p\_{1}\|},0))\,\text{d}t \\ &=\int\_{0}^{||p\_{2}-p\_{1}||}\text{d}F\_{\frac{p\_{2}\cdot p\_{1}}{\|p\_{2}\cdot p\_{1}\|}}\,\frac{p\_{2}-p\_{1}}{||p\_{2}-p\_{1}||}\,\text{d}t,\end{array} \tag{37}$$

where α<sup>0</sup> is the tautological one-form, precisely α<sup>0</sup>ξðXÞ :¼ ξðπ<sup>0</sup>�XÞ for any X ∈ TξT� R2 , and dα<sup>0</sup> ¼ ω0. Applying the basic equality that dFξðξÞ ¼ 1, which is derived from the positive homogeneity of F, for all ξ∈ SR<sup>2</sup> , the above quantity becomes <sup>ð</sup>jjp2−p1jj 0 1dt, which equals jjp2−p1jj.

Let R :¼ fξ<sup>x</sup> ∈S� <sup>R</sup><sup>2</sup> : <sup>x</sup><sup>∈</sup> <sup>p</sup>1p<sup>2</sup> <sup>g</sup> and <sup>T</sup> ¼ fl<sup>∈</sup> Gr1ðR<sup>2</sup><sup>Þ</sup> : <sup>l</sup>∩p1p<sup>2</sup> <sup>≠</sup>Øg, and <sup>p</sup>′ is the projection (composition) from S� <sup>R</sup><sup>2</sup> to Gr1ðR<sup>2</sup> Þ.

Apply the above fact and p′ � ω ¼ ω0,

$$\begin{array}{rclclcl}\int\_{T}|\omega|&=&\int\_{\vec{p'}(\mathbb{R})}|\omega|&=&\int\_{R}|\boldsymbol{p'^\*}\omega|&=&\int\_{R}|\omega\_{0}|\\&&=&|\int\_{R^{+}}\omega\_{0}|+|\int\_{R^{-}}\omega\_{0}|\\&&=&|\int\_{\partial R^{+}}\alpha\_{0}|+|\int\_{\partial R^{-}}\alpha\_{0}|\\&=&4||p\_{2}-p\_{1}|.\end{array}\tag{38}$$

Thus, we have shown the Crofton formula for Minkowski plane.

Furthermore, combining with (33), we have

$$\frac{1}{4} \int\_{l \in Gr\_1(\mathbb{R}^2)} \# (\gamma \mathbb{1} l) |\Omega| = \frac{1}{4} \left[ \underline{\int\_{Gr\_1(\mathbb{R}^2)}} \# (\gamma \mathbb{1} l) |\omega| \right], \tag{39}$$

where Ω ¼ gðθÞdθ∧dr. Then, by the injectivity of cosine transform in Ref. [24], jΩj¼jωj. To obtain the HT area, one can define a map

$$\begin{array}{c} \pi : \overline{Gr\_1(\mathbb{R}^2)} \times \overline{Gr\_1(\mathbb{R}^2)} \wr \dots \to \mathbb{R}^2\\ \pi(l, l') = l \Box', \end{array} \tag{40}$$

extended from Alvarez's construction of taking intersections. The following theorem can be obtained.

Theorem 7.1. For any bounded measurable subset U of a Minkowski plane, we have

$$\text{HT}^2(\mathcal{U}) = \frac{1}{2\pi} \int\_{x \in \mathbb{R}^2} \chi(\mathfrak{x} \cap \mathcal{U}) |\pi\_\* \mathcal{Q}^2|. \tag{41}$$

Proof. On the one hand,

<sup>ϕ</sup><sup>F</sup> : <sup>S</sup>R<sup>2</sup> ! <sup>S</sup>�

ð

ðjjp2−p1jj 0

ðjjp2−p1jj 0

¼

¼

dα<sup>0</sup> ¼ ω0. Applying the basic equality that dFξðξÞ ¼ 1, which is derived from the positive

, the above quantity becomes

ð R jp′ � ωj ¼

#ðγ∩lÞjΩj ¼ <sup>1</sup>

where Ω ¼ gðθÞdθ∧dr. Then, by the injectivity of cosine transform in Ref. [24], jΩj¼jωj.

πðl;l ′ Þ ¼ l∩l ′

<sup>π</sup> : Gr1ðR<sup>2</sup><sup>Þ</sup> · Gr1ðR<sup>2</sup>Þ\<sup>Δ</sup> ! <sup>R</sup><sup>2</sup>

extended from Alvarez's construction of taking intersections. The following theorem can be

4 ð

where α<sup>0</sup> is the tautological one-form, precisely α<sup>0</sup>ξðXÞ :¼ ξðπ<sup>0</sup>�XÞ for any X ∈ TξT�

ð

<sup>L</sup> · { <sup>p</sup>2−p<sup>1</sup> jjp2−p1jj}

ϕ� <sup>F</sup>α<sup>0</sup> ¼

we can obtain a fact that

38 Lagrangian Mechanics

homogeneity of F, for all ξ∈ SR<sup>2</sup>

Apply the above fact and p′

ð T jωj ¼

Furthermore, combining with (33), we have

To obtain the HT area, one can define a map

1 4 ð

<sup>R</sup><sup>2</sup> to Gr1ðR<sup>2</sup>Þ.

� ω ¼ ω0,

> ð p′ ðRÞ

Thus, we have shown the Crofton formula for Minkowski plane.

<sup>l</sup>∈Gr1ðR<sup>2</sup><sup>Þ</sup>

jωj ¼

Let R :¼ fξ<sup>x</sup> ∈S�

position) from S�

obtained.

R2

<sup>ϕ</sup>Fð<sup>L</sup> · { <sup>p</sup>2−p<sup>1</sup> jjp2−p1jj}<sup>Þ</sup>

<sup>ϕ</sup>Fðx;ξÞ¼ðx;dFξÞ, (36)

ðð <sup>p</sup>2−p<sup>1</sup>

<sup>ð</sup> <sup>p</sup>2−p<sup>1</sup> jjp2−p1jj

ðjjp2−p1jj 0

> ω0jþj ð R− ω0j

α0jþj ð ∂R<sup>−</sup> α0j

jjp2−p1jj , <sup>0</sup>ÞÞd<sup>t</sup>

Þdt; (37)

1dt, which equals jjp2−p1jj.

#ðγ∩lÞjωj, (39)

; (40)

R2 , and

(38)

α0

α0d<sup>F</sup> <sup>p</sup>2−p<sup>1</sup> jjp2−p1jj

dF <sup>p</sup>2−p<sup>1</sup> jjp2−p1jj

<sup>R</sup><sup>2</sup> : <sup>x</sup><sup>∈</sup> <sup>p</sup>1p<sup>2</sup> <sup>g</sup> and <sup>T</sup> ¼ fl<sup>∈</sup> Gr1ðR<sup>2</sup><sup>Þ</sup> : <sup>l</sup>∩p1p<sup>2</sup> <sup>≠</sup>Øg, and <sup>p</sup>′ is the projection (com-

ð R jω0j

¼ j ð Rþ

¼ j ð ∂R<sup>þ</sup>

Gr1ðR<sup>2</sup> Þ

¼ 4jjp2−p1jj:

$$\frac{1}{\pi} \frac{1}{\pi} \int\_{D^\*U} \omega\_0^2 = \frac{1}{\pi} \int\_{\partial D^\*U} \omega\_0^2 = \frac{1}{\pi} \int\_{S^\*U} \alpha\_0 \wedge \omega\_0. \tag{42}$$

On the other hand,

1 π ð x∈ R<sup>2</sup> <sup>χ</sup>ðx∩UÞπ�Ω<sup>2</sup> <sup>¼</sup> <sup>1</sup> π ð n ðl;l ′ Þ<sup>∈</sup>Gr1ðR<sup>2</sup> <sup>Þ</sup> ·Gr1ðR<sup>2</sup> Þ\Δ:l∩<sup>l</sup> ′ ∈ U oΩ<sup>2</sup> ¼ <sup>1</sup> π ð fðl;l ′ Þ<sup>∈</sup>Gr1ðR<sup>2</sup> <sup>Þ</sup> ·Gr1ðR<sup>2</sup>Þ\Δ:l∩<sup>l</sup> ′ ∈ Ug ω2 ¼ <sup>1</sup> π ð T�U\fðx;ξ;ξÞ:ξ∈S� <sup>x</sup>Ug p′ � ω2 ¼ <sup>1</sup> π ð T�U\fðx;ξ;ξÞ:ξ∈S� <sup>x</sup>Ug ω2 0 ¼ <sup>2</sup> π ð fðx;ξ;ξÞ:ξ∈S� <sup>x</sup>Ug α0∧ω<sup>0</sup> ¼ <sup>2</sup> π ð S�U α0∧ω0; (43)

where

$$\mathbb{T}^\*\mathcal{U} := \left\{ (\mathbf{x}, \xi, \xi') : \xi, \xi' \in S\_\mathbf{x}^\*\mathcal{U} \right\}. \tag{44}$$

So the claim follows.

Remark 7.2. Lagrangian structure provides the underlying differential structure needed to measure the Holme-Thompson area in integral geometry and therefore is essential and doundamental in integral geometry. For Finsler manifolds, real or complex, it is necessary to analyze the Lagrangian structure of the Finsler manifolds, in the forms of symplectic structure and Kahler structure, and many Finsler manifolds may not have a Lagrangian structure, about which one can refer to Ref. [25]. However, for smooth projective Finsler spaces, the integral geometry formulas have been studied in Ref. [26], for instance.
