I have a question about the first result in the [paper](http://m.mathnet.ru/links/48cdb1d6a1e6777814a8a8b543098968/aa102.pdf) "two
counterexamples in low-dimensional length geometry", by Burago, Ivanov and Shoenthal.


First there is an **open question**: In the two-dimensional disk, can any length
metric can be approximated by Finsler metrics *from below*? Here a length metric is one which is equal to the infimum of curve lengths; it need not be a Riemannian or even a Finsler metric.


The above paper shows that in dimension 3 there is a
counterexample $(D^3\subset \mathbb{E}^3,d)$. The idea of the construction is as follows: There is a
non-trivial shortest geodesic from $a$ to $b$ with $a\_i\rightarrow
a,\ b\_i \rightarrow b$ s.t. $d(a\_i,b\_i)\rightarrow 0$. 


Here I think that $d(a,b)$ may be 0, in which case $d$ is not a metric. I do not understand lemma 2.3, which claims to prove that $d$ is a metric. **What am I missing?**


Thank you for your attention. 


**More detail** (Not question. It is for better understanding on their proof) : 


In $\mathbb{E}^3$ we perturb a tubular neighborhood
$U\_i$ of the segment $a\_ib\_i$ wrt the Euclidean metric $d\_E$. In $U\_i$,
consider linked simple closed curves $\{ S\_k\}\_{k=1}^{n\_i}$ where $a\_i\in S\_1,\ b\_i\in
S\_{n\_i},\ S\_k$ is a simple closed curve and they are linked (Note that this can not happen in dimension 2.)


That is, around $S\_k$ we give a metric $\frac{1}{in\_i} d\_E$, as elaborated in the paper.


To show that $d(a\_i,b\_i)$ is very small, they use the fact :



> 
> (Linked Circles) In $(\mathbb{E}^3,d\_E)$, if two linked simple closed curves lie at distance at least $1$ from each other then length of each curve is at least $2\pi $ 
> 
> 
> 


Finally, it seems to me that they used the fact:



> 
> If a length space $X\_n$ has a Gromov-Hausdorff limit $X$ which is complete, then $X$ is a length space.
> 
> 
> 



Approximating a length metric from below with Finsler metrics
Construct a length metric space as follows : Around given shortest path between two points, construct linked circles s.t. each circle has small length. 


If the length metric is Finsler, then two points around circles has a small distance. 


