  If $\Omega$ is the interior of a convex polygon in $\mathbb{R}^{2}$ and $f,g$
two asymptotic geodesics, we show that the distance function
$d\left(f\left(t\right),g\left(t\right)\right)$ is convex for $t$ sufficiently
large. The same result is obtained in the case $\partial \Omega$ is of class
$C^{2}$ and the curvature of $\partial \Omega$ at the point
$f\left(\infty\right)=g\left(\infty\right) $ does not vanish. An example is
provided for the necessity of the curvature assumption.
