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 Omega in a Banach space such that there is no geodesic between any given pair of points x, y is an element of Omega. In addition, we prove that if 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 X, is smooth.