Integration over convex curves

Question

Let $S$ be a noncompact closed convex proper subset of $\mathbb{C}.$ It is well-known that the boundary $\partial S$ is a rectifiable curve et hence, that we can make integration over it. Is there some easy criterion to check the integrability over such convex curves ?

For example, it seems that for all $S$, we have $$\int_{\partial S} e^{-|w|} dw < \infty,$$ but how can we prove it ? We know that $S$ can be parametrized by $\gamma$ which is absolutely continuous and such that $\lim_{t\to \infty} |\gamma(t)| = +\infty$ but it is not enough to check the integrability.

Answer 1

All that is needed is an estimate of the length of the piece of $\gamma$ in the disk $|z|\leq r$. This can be done as follows.

Remove a bounded arc of $\gamma$ so that two unbounded pieces remain, and the direction of the support line on each piece varies between $a$ and $b$ so that $0<b-a<\pi/2$. We use that this direction is monotone and bounded, which follows from convexity.

It is sufficient to consider one of these unbounded pieces.

Let the beginning of this piece be $z_0$ and let $z$ be any other point on it. Draw the support lines at $z_0$ and $z$; they will intersect at some point $z_1$. Then the angle $(z_0,z_1,z)$ is greater than $\pi/2$. Now the length of your curve from $z_0$ to $z$ is at most $|z_0-z_1|+|z_1-z|\leq 2|z-z_0|$ (draw a picture) and this is all that you need.