Convexity of asymptotic geodesics in Hilbert geometry (Q2094269)

Let \(\Omega\) denote the interior of a convex body \(K \subset \mathbf{R}^n\) equipped with the Hilbert metric \(h\), and let \(f\) and \(g\) be Euclidean asymptotic geodesics on \(\Omega\), which are traces of moving points \(f(t)\) and \(g(t)\), \(-\infty < t < \infty\). The geodesics \(f\) and \(g\) are called asymptotic provided \(f(\infty) = g(\infty)\). The authors prove the existence of a scalar \(T > 0\) such that the distance function \(t \to h(f(t), g(t))\), \(t > T\), is convex provided one of the following sufficient conditions holds: \((a)\) \(K\) is a polytope, \((b)\) the boundary of \(K\) is \(C^2\) and the curvature of \(K\) at the point \(f(\infty) = g(\infty)\) does not vanish. An example is provided for the necessity of the curvature assumption.