  In [12] it has been shown that $(p,q)$ Sobolev inequality with $p>q$ implies
the doubling condition on the underlying measure. We show that even weaker
Orlicz-Sobolev inequalities, where the gain on the left-hand side is smaller
than any power bump, imply doubling. Moreover, we derive a condition on the
quantity that should replace the radius on the righ-hand side (which we call
`superradius'), that is necessary to ensure that the space can support the
Orlicz-Sobolev inequality and simultaneously be non-doubling.
