  In this paper, we prove that if $D\subset R^n$ is a John domain which is
homeomorphic to a uniform domain via a quasiconformal mapping, then each
quasihyperbolic geodesic in $D$ is a cone arc, which shows that the answer to
one of open problems raised by Heinonen in \cite{H} is affirmative. This result
also shows that the answer to the open problem raised by Gehring, Hag and
Martio in \cite{Gm} is positive for John domains which are homeomorphic to
uniform domains via uasiconformal mappings. As an application, we prove that if
$D\subset R^n$ is a John domain which is homeomorphic to a uniform domain, then
$D$ must be a quasihyperbolic $(b, \lambda)$-uniform domain.

Functional Analysis