  Let $\mu$ be a probability measure on $\mathbb{R}^n$ with a bounded density
$f$. We prove that the marginals of $f$ on most subspaces are well-bounded. For
product measures, studied recently by Rudelson and Vershynin, our results show
there is a trade-off between the strength of such bounds and the probability
with which they hold. Our proof rests on new affinely-invariant extremal
inequalities for certain averages of $f$ on the Grassmannian and affine
Grassmannian. These are motivated by Lutwak's dual affine quermassintegrals for
convex sets. We show that key invariance properties of the latter, due to
Grinberg, extend to families of functions. The inequalities we obtain can be
viewed as functional analogues of results due to Busemann--Straus, Grinberg and
Schneider. As an application, we show that without any additional assumptions
on $\mu$, any marginal $\pi_E(\mu)$, or a small perturbation thereof, satisfies
a nearly optimal small-ball probability.
