Generalizing results from
$[\text{MM}1]$
referring to the intersection body
$\text{IK}$
and the cross-section body
$CK$
of a convex body
$K\,\,\subset \,\,{{\mathbb{R}}^{d}}$
,
$d\,\ge \,2$
, we prove theorems about maximal
$k$
-sections of convex bodies,
$k\,\in \,\{1,\,\ldots ,\,d\,-\,1\}$
, and, simultaneously, statements about common maximal
$(d\,-\,1)-$
and 1-transversals of families of convex bodies.