Intrinsic geometry and analysis of Finsler structures

Abstract

In this short note, we prove that if F is a weak upper semicontinuous admissible Finsler structure on a domain in $\mathbb {R}^n, n\ge 2$, then the intrinsic distance and differential structures coincide.

1 Introduction

Let $\varOmega \subset \mathbb {R}^n, n\ge 2$, be a domain and F an admissible Finsler structure on $\varOmega$ (the precise definition is given in Sect. 2 below). Associated with F, we have the following intrinsic distance defined by

$$\begin{aligned} \delta _F(x,y)=\sup _{u}\left\{ u(x)-u(y): u \text { is Lipschitz and } \Vert F(x, \hbox {d}u(x))\Vert _\infty \le 1\right\} {.} \end{aligned}$$

(1.1)

Above, $\hbox {d}u(x)$ denotes the differential of the Lipschitz function u at a point x. Recall that the well-known Rademacher’s theorem implies that $\hbox {d}u(x)$ exists at almost every $x\in \varOmega$, and thus the above definition makes sense. The ellipticity condition on F implies that $\delta _F$ is locally comparable to the standard Euclidean distance. We define the pointwise Lipschitz constant of a Lipschitz function $u:\varOmega \rightarrow \mathbb {R}$ by setting

$$\begin{aligned} {{\mathrm{Lip}}}_{\delta _F}u(x)=\limsup _{y\rightarrow x}\frac{|u(y)-u(x)|}{\delta _F(x,y)}. \end{aligned}$$

Given a subset K of $\mathbb {R}^n$, we set

$$\begin{aligned} {{\mathrm{Lip}}}_{\delta _F}(u,K)=\sup _{x,y\in K,x\ne y}\frac{|u(x)-u(y)|}{\delta _F(x,y)} \end{aligned}$$

and denote by ${{\mathrm{Lip}}}_{\delta _F}(K)$ the collection of all functions $u:K\rightarrow \mathbb {R}$ with ${{\mathrm{Lip}}}_{\delta _F}(u,K)<\infty$.

Sturm asked the following interesting question in [12]: Is a diffusion process determined by the intrinsic distance? Mathematically, Sturm’s question can be formulated as follows: Is it true that for all $u\in {{\mathrm{Lip}}}_{\delta _F}(\varOmega )$,

$$\begin{aligned} F(x, \hbox {d}u(x))={{\mathrm{Lip}}}_{\delta _F}u(x) \end{aligned}$$

almost everywhere with $F(x,v)=\sqrt{\langle A(x)v,v\rangle }$?

The answer to the question is yes when A is supposed to be continuous, as shown by Sturm [12, Proposition 4]. He also pointed out that the answer to this question is not always positive [12, Theorem 2]: For $F(x,v)=\sqrt{\langle A(x)v,v\rangle }$, where A is a diffusion matrix, there exists $\tilde{F}(x,v)= \sqrt{\langle \tilde{A}(x)v,v\rangle }$ such that $\delta _F=\delta _{\tilde{F}}$ but

$$\begin{aligned} F(x,v)<\tilde{F}(x,v) \end{aligned}$$

for all $v\in \mathbb {R}^n{\backslash } \{0\}$; see also [11] for a different example.

The case $F(x,v)=\sqrt{\langle A(x)v,v\rangle }$ gained deeper understanding in a recent paper [10], where the authors enhanced Sturm’s result by showing that if the diffusion matrix A is weak upper semicontinuous, then the differential and distance structures coincide. They also constructed an example, which shows that if A fails to be upper semicontinuous on a set of positive measure, then the differential and distance structure may fail to coincide.

The main purpose of this paper is to generalize the above result of [10] to more general Finsler structures. More precisely, we are going to prove the following result.

Theorem 1.1

Let $n\ge 2$ and F be an admissible Finsler structure on a domain $\varOmega \subset \mathbb {R}^n$. If F is weak upper semicontinuous on $\varOmega$, then the intrinsic distance and differential structure coincide. That is given a Lipschitz function u on $\varOmega$ (with respect to the Euclidean distance), for almost every $x\in \varOmega$, we have

$$\begin{aligned} {{\mathrm{Lip}}}_{\delta _F}u(x)=F(x, \mathrm{d}u(x)). \end{aligned}$$

The proof of [10, Theorem 2] relies heavily on the structure of $F(x,v)=\sqrt{\langle A(x)v,v\rangle }$. It seems that there is little hope to adapt their proofs in the greater generality of this paper.

To see an example where Theorem 1.1 applies more generally than [10, Theorem 2], we may choose suitable weighted $L^p$-norm with $1\le p<\infty$. For instance, consider $F(x,v)=(\sum \nolimits _{i=1}^n w(x)|v_i|^p)^{1/p}$, where the weight function w is upper semicontinuous and satisfies the ellipticity condition $0<c\le w(x)\le C<\infty$ for all $x\in \mathbb {R}^n$.

Theorem 1.1 can be regarded as an improved version of [8, Proposition 2.4] from $L^\infty$-norm to pointwise equality.

Our proof of Theorem 1.1 completely differs from that used in [10] and it is simpler than [10], even in their setting. The crucial observation is Proposition 3.1 below, a special case of a result due to De Cecco and Palmieri [6], which states that the intrinsic distance $\delta _\mathrm{F}$ (infinitesimally) coincides with $d_\mathrm{c}^*$, where $d_\mathrm{c}^*$ is the cc-distance induced by the Finsler structure F. The weak upper semicontinuity is crucial for our proof, since it implies that the “metric density” of a curve with respect to the metric length coincides with its “differential density”; see Sect. 4 below for the precise meaning. Our approach is more geometric and was influenced a lot by the recent studies in Finsler geometry [2, 4, 6, 7]. Some of the ideas from this paper were successfully used in our companion paper [9] on certain $L^\infty$-variational problems associated with measurable Finsler structures. It is known (e.g., [1, 11]) that the intrinsic distance and differential structures coincide even for abstract Dirichlet forms on metric measure spaces. It would be interesting to know that whether a version of Theorem 1.1 holds in the abstract setting as there.

This paper is organized as follows. Section 2 contains all the preliminaries related to Finsler structures. Sections 3 and 4 contain an overview of the necessary background that are needed for our proof of Theorem 1.1. In Sect. 5, we prove Theorem 1.1. “Appendix” contains a separate proof of Proposition 3.1 under the weak upper semicontinuity assumption.

2 Preliminaries on Finsler structures

Let $\varOmega \subset \mathbb {R}^n, n\ge 2$, be a domain, i.e., an open connected set.

Definition 2.1

(Finsler structures) We say that a function $F:\varOmega \times \mathbb {R}^n\rightarrow [0,\infty )$ is a Finsler structure on $\varOmega$ if

• $F(\cdot ,v)$ is Borel measurable for all $v\in \mathbb {R}^n, F(x,\cdot )$ is continuous for a.e. $x\in \varOmega$;

• $F(x,v)>0$ for a.e. x if $v\ne 0$;

• $F(x,\lambda v)=|\lambda | F(x,v)$ for a.e. $x\in \varOmega$ and for all $\lambda \in \mathbb {R}$ and $v\in \mathbb {R}^n$.

Definition 2.2

(Admissible Finsler structures) A Finsler structure F is said to be admissible if

• $F(x,\cdot )$ is convex for a.e. $x\in \varOmega$;

• F is locally equivalent to the Euclidean norm or elliptic, i.e., there exists a continuous function $\lambda :\varOmega \rightarrow [1,\infty )$ such that

  $$\begin{aligned} \frac{1}{\lambda (x)}|v|\le F(x,v)\le \lambda (x)|v| \end{aligned}$$

  for a.e. $x\in \varOmega$ and for all $v\in \mathbb {R}^n$.

It is straightforward to verify that the standard $L^p$-norm ($1\le p<\infty$), i.e., $F(x,v)=(\sum \nolimits _{i=1}^nv_i^p)^{1/p}$, is an admissible Finsler structure on $\mathbb {R}^n$. From the geometric point of view, there are many other interesting examples and we refer the interested readers to [2] for the details.

Recall that a function $u:\varOmega \rightarrow \mathbb {R}$ is said to be upper semicontinuous at $x\in \varOmega$ if

$$\begin{aligned} u(x)\ge \limsup _{y\rightarrow x}u(y). \end{aligned}$$

Following [10], we say that u is weak upper semicontinuous in $\varOmega$ if u is upper semicontinuous at almost every $x\in \varOmega$. Let F be an admissible Finsler structure on $\varOmega$. We say that F is weak upper semicontinuous on $\varOmega$ if for each $v\in \mathbb {R}^n$, the function $F(\cdot ,v)$ is weak upper semicontinuous on $\varOmega$.

Similarly a function $u:\varOmega \rightarrow \mathbb {R}$ is said to be lower semicontinuous at $x\in \varOmega$ if

$$\begin{aligned} u(x)\le \liminf _{y\rightarrow x}u(y), \end{aligned}$$

and u is weak lower semicontinuous in $\varOmega$ if u is lower semicontinuous at almost every $x\in \varOmega$. Let F be an admissible Finsler structure on $\varOmega$. We say that F is weak lower semicontinuous on $\varOmega$ if for each $v\in \mathbb {R}^n$, the function $F(\cdot ,v)$ is weak lower semicontinuous on $\varOmega$.

Let F be an admissible Finsler structure for $\varOmega$. We introduce the dual of $F:\varOmega \times \mathbb {R}^n\rightarrow [0,\infty )$ in the standard way.

Definition 2.3

(Dual Finsler structures) The dual $F^*$ of an admissible Finsler structure $F:\varOmega \times \mathbb {R}^n\rightarrow [0,\infty )$ is defined as

$$\begin{aligned} F^*(x,w)&=\sup _{v\in \mathbb {R}^n}\left\{ \langle v,w\rangle : F(x,v)\le 1\right\} \\&=\max _{v\ne 0}\left\{ \left\langle w,\frac{v}{F(x,v)}\right\rangle \right\} , \end{aligned}$$

where $\langle \cdot ,\cdot \rangle$ is the standard inner product in $\mathbb {R}^n$.

The following proposition follows immediately from Definition 2.3; see for instance [8, Section 1.2] or [3, Section 2] for more information.

Proposition 2.4

(Basic properties of a dual Finsler structure) Let F be an admissible Finsler structure on $\varOmega$. Then the dual function $F^*$ satisfies the following properties

• $F^*(\cdot ,v)$ is Borel measurable and $F^*(x,\cdot )$ is Lipschitz;

• $F^*(x,\cdot )$ is a norm;

• $F^*(x,\cdot )$ is locally equivalent to the Euclidean norm, i.e.

  $$\begin{aligned} \frac{1}{\lambda (x)}|v|\le F^*(x,v)\le \lambda (x)|v|. \end{aligned}$$

• $(F^*)^*(x,v)= F(x,v)$;

• F is weak upper (lower) semicontinuous if and only if $F^*$ is weak lower (upper) semicontinuous.

3 Comparison of intrinsic distances

Let $(\varOmega ,F(x,\cdot ),d_\mathrm{c}^F,\delta _F)$ be a Finsler manifold with an admissible Finsler structure F. For an admissible Finsler structure F on $\varOmega$, we may associate a cc-distance in the standard way by setting

$$\begin{aligned} d_\mathrm{c}^*(x,y)&:=\sup _{N}\inf _{\gamma \in \Gamma _N^{x,y}}\left\{ \int _0^1 F^*\left( \gamma (t),\gamma ^{\prime }(t)\right) \hbox {d}t\right\} , \end{aligned}$$

where the supremum is taken over all subsets N of $\varOmega$ such that $|N|=0$ and $\Gamma _N^{x,y}(\varOmega )$ denotes the set of all Lipschitz curves in $\varOmega$ with end points x and y transversal to N, i.e., $\mathscr {H}^1(N\cap \gamma )=0$. For an admissible Finsler metric $F, d_\mathrm{c}^*$ is indeed an intrinsic distance; for the definition of an intrinsic distance and this fact, see [6, 7]. Above, we use |E| to denote the n-dimensional Lebesgue measure of a set $E\subset \mathbb {R}^n$ and $\mathcal {H}^1$ the one-dimensional Hausdorff measure.

The following fundamental result, which relates $\delta _F$ and $d_\mathrm{c}^*$, was a special case of [6, Theorem 3.7].

Proposition 3.1

Let F be an admissible Finsler structure on $\varOmega$. Then for almost every $x\in \varOmega$, it holds

$$\begin{aligned} \lim _{y\rightarrow x}\frac{\delta _F(x,y)}{d_\mathrm{c}^*(x,y)}=1. \end{aligned}$$

(3.1)

Since we have assumed the weak upper semicontinuity on our admissible Finsler structure in our main result Theorem 1.1, we give a separate proof of Proposition 3.1 under this extra assumption in “Appendix.”

4 Comparison of metric derivatives

For any distance d on $\varOmega$ and each Lipschitz (with respect to d) curve $\gamma :[a,b]\rightarrow \varOmega$, the length of $\gamma$ with respect to d is denoted by $\mathcal {L}_d(\gamma )$, i.e.,

$$\begin{aligned} \mathcal {L}_d(\gamma ):=\sup \left\{ \sum _{i=1}^k d(\gamma (t_i),\gamma (t_{i+1})) \right\} , \end{aligned}$$

where the supremum is taken over all partitions $\{[t_i,t_{i+1}]\}$ of [a, b].

Given a curve $\gamma$, the metric derivative of $\gamma$ at t is defined to be

$$\begin{aligned} \left| \gamma ^{\prime }(t)\right| _d:=\limsup _{s\rightarrow 0}\frac{d(\gamma (t+s),\gamma (t))}{s}. \end{aligned}$$

If $\gamma :[a,b]\rightarrow \varOmega$ is Lipschitz with respect to d, then its length can be computed by integrating the metric derivative, i.e.

$$\begin{aligned} \mathcal {L}_d(\gamma )=\int _a^b|\gamma ^{\prime }(t)|_d \hbox {d}t. \end{aligned}$$

In other words, for a Lipschitz curve, the metric derivative is the metric density of its length.

For any intrinsic distance d, which is locally bi-Lipschitz equivalent to the Euclidean distance, we may associate a Finsler structure $\varDelta _d$ in the following manner. For each $x\in \varOmega$ and for every direction v, we define

$$\begin{aligned} \varDelta _d(x,v):=\limsup _{t\rightarrow 0^+}\frac{d(x,x+tv)}{t}. \end{aligned}$$

(4.1)

It can be proved that for every Lipschitz curve $\gamma :[a,b]\rightarrow \varOmega$, we have

$$\begin{aligned} \mathcal {L}_d(\gamma )=\int _a^b\varDelta _d\left( \gamma (t),\gamma ^{\prime }(t)\right) \hbox {d}t. \end{aligned}$$

In particular, $\varDelta _d(\gamma (t),\gamma ^{\prime }(t))=|\gamma ^{\prime }(t)|_{d}$ for a.e. $t\in [a,b]$.

Remark 4.1

For any admissible Finsler structure F, one always has

$$\begin{aligned} \varDelta _{d_\mathrm{c}^*}(x,v)\le F^*(x,v)\quad \text { for a.e. }\,x\in \varOmega \quad \text { and all}\, v\in \mathbb {R}^n; \end{aligned}$$

(4.2)

see [8, Proposition 1.6]. However, the equality does not necessary hold; see [7, Example 5.1] for a counterexample.

In addition, for an admissible Finsler structure F, the dual Finsler structure $F^*$ always induces a lower semicontinuous length structure; see [4, Section 2.4.2]. Moreover, if the Finsler metric F is weak upper semicontinuous on $\varOmega$, then the following stronger result holds.

Proposition 4.2

([3, Proposition 2.9]) If the Finsler structure F is weak upper semicontinuous on $\varOmega$, then for a.e. $x\in \varOmega$ and all $v\in \mathbb {R}^n$, it holds

$$\begin{aligned} \varDelta _{d_\mathrm{c}^*}(x,v)=F^*(x,v). \end{aligned}$$

5 Coincidence of distance structure and differential structure

In this section, we are ready to prove our main result Theorem 1.1.

Proposition 5.1

For each $u\in {{\mathrm{Lip}}}_{\delta _F}(\varOmega ), F(x, \mathrm{d}u(x))\le {{\mathrm{Lip}}}_{\delta _F}u(x)$ for a.e. $x\in \varOmega$.

Proof

Since both sides are positively 1-homogeneous with respect to u, we only need to show that for a.e. $x\in \varOmega$, if ${{\mathrm{Lip}}}_{\delta _F}u(x)= 1$, then $F(x, \hbox {d}u(x))\le 1$.

Note that by Proposition 3.1, for a.e. $x\in \varOmega , {{\mathrm{Lip}}}_{\delta _F}u(x)={{\mathrm{Lip}}}_{d_\mathrm{c}^*}u(x)$. Fix such an x. For each $v\in \mathbb {R}^n$, we have

$$\begin{aligned} \hbox {d}u(x)v&=\lim _{t\rightarrow 0}\frac{u(x+tv)-u(x)}{t}\\&\le \limsup _{t\rightarrow 0}\frac{d_\mathrm{c}^*(x,x+tv)}{t}\cdot \limsup _{t\rightarrow 0}\frac{u(x+tv)-u(x)}{d_\mathrm{c}^*(x,x+tv)}\\&\le \varDelta _{d_\mathrm{c}^*}(x,v){{\mathrm{Lip}}}_{d_\mathrm{c}^*}u(x)\le F^*(x,v), \end{aligned}$$

where in the last inequality, we have used the inequality (4.2).

Therefore,

$$\begin{aligned} F(x, \hbox {d}u(x))&=F^{**}(x,\hbox {d}u(x))\\&=\max _{v\ne 0} \left\{ \hbox {d}u(x)\left( \frac{v}{F^*(x,v)}\right) \right\} \le 1 \end{aligned}$$

as desired. This completes our proof.

Theorem 5.2

Let F be an admissible Finsler structure on $\varOmega$. If F is weak upper semicontinuous on $\varOmega$, then for any Lipschitz function u in $(\varOmega ,\delta _F)$,

$$\begin{aligned} {{\mathrm{Lip}}}_{\delta _F}u(x)\le F(x, \mathrm{d}u(x)) \end{aligned}$$

for a.e. $x\in \varOmega$.

Proof

First, note that our assumption on F implies that F satisfies the following uniform upper semicontinuity property, for a.e. $x\in \varOmega$,

$$\begin{aligned} \forall \varepsilon>0, \quad \exists \delta >0: F(y,v)\le (1+\varepsilon ) F(x,v)\quad \text {for all }\,y\in B(x,\delta ),\quad v\in \mathbb {R}^n. \end{aligned}$$

(5.1)

By homogeneity of F (with respect to v), it suffices to prove (5.1) for all $v\in \mathbb {S}$ (the unit sphere). Suppose by contradiction, that (5.1) fails. Then there exist some $x\in \varOmega$ and some $\varepsilon _0>0$ such that for each $k\in \mathbb {N}$, there exist some $y_k\in B\left( x,\frac{1}{k}\right)$ and $v_k\in \mathbb {S}$ so that

$$\begin{aligned} F(y_k,v_k)>(1+\varepsilon _0)F(x,v_k). \end{aligned}$$

(5.2)

By compactness of $\mathbb {S}$, we may assume (up to another subsequence if necessary) $v_k\rightarrow v\in \mathbb {S}$ as $k\rightarrow \infty$. Then

$$\begin{aligned} F(x,v)&=\limsup _{k\rightarrow \infty }F(x,v_k)\ge \limsup _{k\rightarrow \infty }\limsup _{y\rightarrow x}F(y,v_k)\\&\ge \limsup _{k\rightarrow \infty }F(y_k,v_k)\ge \limsup _{k\rightarrow \infty }(1+\varepsilon _0)F(x,v_k)\\&=(1+\varepsilon _0)F(x,v), \end{aligned}$$

which is a contradiction.

Secondly, by Rademacher’s theorem, it suffices to prove Theorem 5.2 when $u(x)=\langle v,x\rangle$ is linear. We may additionally assume that $v\ne 0$. By the fundamental theorem of calculus and the definition of $F^*$, we have

$$\begin{aligned} |u(x)-u(y)|&=|\langle v,y-x\rangle |=\left| \int _0^1 \frac{\hbox {d}}{\hbox {d}t}u(\gamma (t))\hbox {d}t\right| \\&=\left| \int _0^1\langle v,\gamma ^{\prime }(t)\rangle \hbox {d}t\right| \le (1+\varepsilon )F(x,v)\int _0^1 F^*\left( \gamma (t),\gamma ^{\prime }(t)\right) \hbox {d}t \end{aligned}$$

whenever x, y and $\gamma (t)$ belongs to the “$\delta$-neighborhood of x where (5.1) holds; it follows that

$$\begin{aligned} \frac{|\langle v,y-x\rangle |}{d_\mathrm{c}^*(x,y)}\le (1+\varepsilon )F(x,v), \end{aligned}$$

whenever $|x-y|<\delta$. Letting $y\rightarrow x$ and $\varepsilon \rightarrow 0$ concludes our proof. $\square$

Appendix: Proof of Proposition 3.1 when F is weak upper semicontinuous

Proof

The inequality $\delta _F(x,y)\le d_\mathrm{c}^*(x,y)$ follows directly from definitions. Indeed, for each Lipschitz function u with $\Vert F(\cdot , \hbox {d}u(\cdot ))\Vert _{L^{\infty }(\varOmega )}\le 1$, each $x,y\in \varOmega$, for each Lipschitz curve $\gamma$ joining x and y that is transversal to the zero measure set $N=\{x\in \varOmega :F(x, \hbox {d}u(x))> 1\}$,

$$\begin{aligned} u(x)-u(y)&=\int _0^1 \hbox {d}u(\gamma (t))\left( \gamma ^{\prime }(t)\right) \hbox {d}t\\&\le \int _0^1 F^*\left( \gamma (t),\gamma ^{\prime }(t)\right) \hbox {d}t=\mathcal {L}_{d_\mathrm{c}^*}(\gamma ), \end{aligned}$$

where $\mathcal {L}_{d_\mathrm{c}^*}$ denotes the length of the curve $\gamma$ with respect to the metric $d_\mathrm{c}^*$. Taking infimum over all admissible curves on the right-hand side and then supermum over all admissible functions over the left-hand side, we obtain via Proposition 4.2 that

$$\begin{aligned} \delta _F(x,y)\le d_\mathrm{c}^*(x,y). \end{aligned}$$

In particular,

$$\begin{aligned} \limsup _{y\rightarrow x}\frac{\delta _F(x,y)}{d_\mathrm{c}^*(x,y)}\le 1. \end{aligned}$$

We are left to prove that

$$\begin{aligned} \liminf _{y\rightarrow x}\frac{\delta _F(x,y)}{d_\mathrm{c}^*(x,y)}\ge 1. \end{aligned}$$

(5.3)

We divide the proof of this equation into two steps.

Step 1 Assume that $F(\cdot ,v)$ is continuous.

Fix $x\in \varOmega$ and $\varepsilon >0$. Since $F(\cdot ,v)$ and $F^*(\cdot ,v)$ are continuous in $B(x,\delta )$, we may assume that for all $z\in B(x,\delta )$,

$$\begin{aligned} (1-\varepsilon )F(z,v)\le F(x,v)\le (1+\varepsilon )F(z,v) \end{aligned}$$

and

$$\begin{aligned} (1-\varepsilon )F^*(z,v)\le F^*(x,v)\le (1+\varepsilon )F^*(z,v). \end{aligned}$$

Note that the issue is local, we are now restricting ourselves to the ball $B(x,\delta )$.

Consider the curve $\gamma (t)=x+t(y-x)$, we have

$$\begin{aligned} d_{\mathrm{c}}^*(x,y)\le \mathcal {L}_{d_\mathrm{c}^*}(\gamma )=\int _0^1F^*\left( \gamma (t),\gamma ^{\prime }(t)\right) \hbox {d}t\le (1+\varepsilon )F^*(x,y-x). \end{aligned}$$

By the definition of a dual Finsler structure, we know that there exists some $\tilde{v}\ne 0$ such that $F^*(x,y-x)=\langle y-x,\frac{\tilde{v}}{F(x,\tilde{v})}\rangle$. Set

$$\begin{aligned} v:=\frac{\tilde{v}}{(1+\varepsilon )F(x,\tilde{v})}. \end{aligned}$$

Then $F(x,v)=\frac{1}{1+\varepsilon }$ and $\langle v,y-x\rangle =\frac{1}{1+\varepsilon }F^*(x,y-x)$. Note that for all $z\in B(x,\delta ), F(z,v)\le (1+\varepsilon )F(x,v)\le 1$ and so the function $u(z):=\langle v,z\rangle$ is an admissible function for $\delta _F(x,y)$. This means that

$$\begin{aligned} \delta _F(x,y)\ge u(y)-u(x)=1{/}(1+\varepsilon )F^*(x,y-x)\ge \frac{1}{(1+\varepsilon )^2}d_\mathrm{c}^*(x,y). \end{aligned}$$

It is clear that (5.3) follows from the above inequality by letting $\varepsilon \rightarrow 0$.

Step 2 Assume that $F(\cdot ,v)$ is weak upper semicontinuous.

In this case, $F^*$ is weak lower semicontinuous, it is a well-known fact that there exists a sequence of admissible Finsler norms $F_n^*(\cdot ,v)$, which is continuous in the first variable, such that

$$\begin{aligned} F_n(x,v)^*\le F_{n+1}^*(x,v)\le \cdots \rightarrow F^*(x,v); \end{aligned}$$

and $d_\mathrm{c}^{*n}\rightarrow d_\mathrm{c}^*$ as $n\rightarrow \infty$, where $d_\mathrm{c}^{*n}$ is the cc-distance induced by the Finsler structure $F_n$; see for instance [5, Section 4]. Let $F_n=F_n^{**}$ denote the dual of $F_n^*$, then it is easy to check from our definition that

$$\begin{aligned} F_n(x,v)\ge F_{n+1}(x,v)\ge \cdots \rightarrow F(x,v). \end{aligned}$$

It follows that

$$\begin{aligned} \frac{\delta _F(x,y)}{d_\mathrm{c}^*(x,y)}=\lim _{n\rightarrow \infty }\frac{\delta _{F_n}(x,y)}{d_\mathrm{c}^{*n}(x,y)}, \end{aligned}$$

where $\delta _{F_n}$ is the intrinsic distance induced by $F_n$ similar as $\delta _{F}$. Given $\varepsilon >0$, there exists $N_0$ such that for all $n\ge N_0$,

$$\begin{aligned} \frac{\delta _F(x,y)}{d_\mathrm{c}^*(x,y)}\ge (1-\varepsilon )\frac{\delta _{F_n}(x,y)}{d_\mathrm{c}^{*n}(x,y)}. \end{aligned}$$

On the other hand, by step 1,

$$\begin{aligned} \liminf _{y\rightarrow x}\frac{\delta _{F_n}(x,y)}{d_\mathrm{c}^{*n}(x,y)}\ge 1. \end{aligned}$$

We thus obtain

$$\begin{aligned} \liminf _{y\rightarrow x}\frac{\delta _{F_n}(x,y)}{d_\mathrm{c}^{*n}(x,y)}\ge \liminf _{y\rightarrow x}(1-\varepsilon )\frac{\delta _{F_n}(x,y)}{d_\mathrm{c}^{*n}(x,y)}\ge 1-\varepsilon . \end{aligned}$$

The claim follows by letting $\varepsilon \rightarrow 0$. $\square$