# Convexity of asymptotic geodesics in Hilbert Geometry

21 Mar 2020 Charitos Charalampos Papadoperakis Ioannis Tsapogas Georgios

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... (read more)

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