  It is shown that each monotone Minkowski endomorphism of convex bodies gives
rise to an isoperimetric inequality which directly implies the classical
Urysohn inequality. Among this large family of new inequalities, the only
affine invariant one - the Blaschke-Santal\'o inequality - turns out to be the
strongest one. A further extension of these inequalities to merely weakly
monotone Minkowski endomorphisms is proven to be impossible. Moreover, for
functional analogues of monotone Minkowski endomorphisms, a family of analytic
inequalities for log-concave functions is established which generalizes the
functional Blaschke-Santal\'o inequality.
