  We give an alternative characterization of the class of Muckenhoupt weights
$A_{\infty, \mathfrak B}$ for homothecy invariant Muckenhoupt bases $\mathfrak
B$ consisting of convex sets. In particular we show that $w\in A_{\infty,
\mathfrak B}$ if and only if there exists a constant $c>0$ such that for all
measurable sets $E\subset \mathbb R^n$ we have $$ w({x\in \mathbb R^n:
M_{\mathfrak B} (\mathbf {1}_E)(x)>1/2}) < c w(E).$$ This applies for example
to the collection $\mathfrak R$ of rectangles with sides parallel to the
coordinate axes, giving a new characterization of strong (multiparameter)
Muckenhoupt weights. We also show versions of these results under the presence
of a doubling measure. Thus the strong maximal function $M_{\mathfrak R,\mu}$,
defined with respect to a product-doubling measure $\mu$, is bounded on
$L^p(\mu)$ for some $p>1$ if and only if $$\mu({x\in \mathbb R^n: M_{\mathfrak
R,\mu} (\mathbf{1}_E)(x)>1/2}) < c \mu(E)$$ for all measurable sets $E\subset
\mathbb R^n$. Finally we discuss applications in differentiation theory,
proving among other things that Tauberian conditions as above imply that the
corresponding bases differentiate $L^\infty(\mu)$, with respect to the measure
$\mu$.
