On Quasihyperbolic Geodesics in Banach Spaces

We study properties of quasihyperbolic geodesics on Banach spaces. For example, we show that in a strictly convex Banach space with the Radon-Nikodym property, the quasihyperbolic geodesics are unique. We also give an example of a convex domain $\mathrm{\Omega }$ in a Banach space such that there is no geodesic between any given pair of points $x,y\in \mathrm{\Omega }\mathrm{.}$ In addition, we prove that if $\mathrm{X}$ is a uniformly convex Banach space and its modulus of convexity is of a power type, then every geodesic of the quasihyperbolic metric, defined on a proper subdomain of $\mathrm{X}$, is smooth.