Approximating a length metric from below with Finsler metrics

I have a question about the first result in the paper "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?

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.

• The statement in your first grey box looks to me like nonsense (why can't $c_1$ and $c_2$ be arbitrarily small?), but I don't see this used in the paper. They use some lemmas that relate the distance and length of LINKED curves in $\mathbb{R}^3$. The second grey box (limit of length spaces is a length space) is a standard fact. For example it is Theorem 7.5.1 in the Burago-Burago-Ivanov book "A Course in Metric Geometry". For the first part of your question, I think you may have gotten confused between the metric $d$ (which is a metric) and the supposed Finsler metric $\rho$. – user103319 Feb 16 '17 at 21:28
• @ Matt F. : Thank you for elaborate editting. – Hee Kwon Lee Feb 17 '17 at 2:55
• @ user103319 : Thank you for your comment. I edited for clarifying – Hee Kwon Lee Feb 17 '17 at 3:00