  Let $\Omega$ be a bounded domain in $\mathbb{R}^n$ with $n\ge2$ and
$s\in(0,1)$. Assume that $\phi : [0, \infty) \to [0, \infty)$ be a Young
function obeying the doubling condition with the constant
$K_\phi<2^{\frac{n}{s}}$. We demonstrate that $\Omega $ supports a
$(\phi_\frac{n}{s}, \phi)$-Poincar\'e inequality if it is is a John domain.
  Alternately, assume further that $\Omega$ is a bounded domain that is
quasiconformally equivalent to some uniform domain when $n\ge3$ or a simply
connected domain when $n=2$. We demonstrate $\Omega$ is a John domain if a
$(\phi_\frac{n}{s}, \phi)$-Poincar\'e inequality holds.
