  Let $\Omega\subset\mathbb R^{n+1}$ be an open set with $n$-AD-regular
boundary. In this paper we prove that if the harmonic measure for $\Omega$
satisfies the so-called weak-$A_\infty$ condition, then $\Omega$ satisfies a
suitable connectivity condition, namely the weak local John condition. Together
with other previous results by Hofmann and Martell, this implies that the
weak-$A_\infty$ condition for harmonic measure holds if and only if
$\partial\Omega$ is uniformly $n$-rectifiable and the weak local John condition
is satisfied. This yields the first geometric characterization of the
weak-$A_\infty$ condition for harmonic measure, which is important because of
its connection with the Dirichlet problem for the Laplace equation.
