  We prove that for a domain $\Omega \subset \mathbb{R}^n$, being
$(\epsilon,\delta)$ in the sense of Jones is equivalent to being an extension
domain for bmo$(\Omega)$, the nonhonomogeneous version of the space of function
of bounded mean oscillation on $\Omega$. In the process we demonstrate that
these conditions are equivalent to local versions of two other conditions
characterizing uniform domains, one involving the presence of length cigars
between nearby points and the other a local version of the quasi-hyperbolic
uniform condition. Our results show that the definition of bmo$(\Omega)$ is
closely connected to the geometry of the domain.
