  We investigate properties of quasihyperbolic balls and geodesics in Euclidean
and Banach spaces. Our main result is that in uniformly smooth Banach spaces a
quasihyperbolic ball of a convex domain is $C^1$-smooth. The question about the
smoothness of quasihyperbolic balls is old, originating back to the discussions
of F.W. Gehring and M. Vuorinen in 1970's. To our belief, the result is new
also in the Euclidean setting. We also address some other issues involving the
smoothness of quasihyperbolic balls.
  We introduce an interesting application of quasihyperbolic metrics to
renormings of Banach spaces. To provide a useful tool for this approach we turn
our attention to the variational stability of quasihyperbolic geodesics.
Several examples and illustrations are provided.
