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Convexity of asymptotic geodesics in Hilbert Geometry

If $Ω$ 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 $\partial Ω$ is of class $C^{2}$ and the curvature of $\partial Ω$ 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.