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A curve $\gamma$ is said rectifiable if its lenght is finite, $L(\gamma) < \infty$ . We say that a compact set $X \subset \mathbb C$ is connected by rectifiable arcs if any two points of $X$ can be joined by a rectiable curve lying within X.

For a set $X$ like above, we can define

$$ \delta (z_1, z_2) = \inf \left \{ L(\gamma): \, \gamma \text{ is a curvejoining } z_1 \text{ and } z_2\right \} $$

which is the geodesic distance between $z_1$ and $z_2$. My questions are:

  1. Why is the infimum attained?

I can't see it just because $X$ is compact.

  1. Showing that $\delta$ defines a metric in $X$.

It's not dificult to proof see $\delta (z_1,z_1) = 0$ and $\delta (z_1,z_2) = \delta (z_2, z_1)$. I couldn't proof that $\delta(z_1, z_3) \leq \delta(z_1, z_2) + \delta (z_2, z_3).$

Help?

1 Answers 1

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1) is not at all trivial, there is a bit of theory behind this, unless during the last 15 years new approaches to this have been found.

This is usually shown in books about the calculus of variations or differential geometry. The problem is that, if a sequence of curves lets the length functional tend to the infimum, then it's not at all clear how to show that the sequence has a subsequence which converges in a reasonable topology. If that were true you were almost done since the length funcional is lower semicontinuous.

One solution is to look at the energy functional $c\mapsto \frac{1}{2}\int_I |c^\prime(t)|^2 dt$ instead. It can be shown that this majorizes the length functional and that a minimum of the energy is also a minimum of the length.

The energy functional is naturally defined on a Sobolev space $H^{1,2}(I)$, and a sequence which has bounded energy has bounded norm in that space, so it will contain a weakly convergent subsequence. Then, with standard arguments from the calculus of variations you can show that the weak limit is a (smooth, if the domain admits this) curve which minimizes energy (hence length) when the set on which you are working is compact.

2), on the other hand, is rather trivial. If you have a curve $c_1$ from $z_1$ to $z_2$ and another curve $c_2$ from $z_2$ to $z_3$ then you get an obvious curve from $z_1$ to $z_3$ the lengths of which is just the sum of the lengths of $c_1$ and $c_2$. But of course you have more choices to join $z_1$ and $z_3$, since there is no need to visit $z_2$. Taking infima then will easily show the result.