  We discuss optimal constants in a recent result of Rudelson and Vershynin on
marginal densities. We show that if $f$ is a probability density on $\R^n$ of
the form $f(x)=\prod_{i=1}^n f_i(x_i)$, where each $f_i$ is a density on $\R$,
say bounded by one, then the density of any marginal $\pi_E(f)$ is bounded by
$2^{k/2}$, where $k$ is the dimension of $E$. The proof relies on an adaptation
of Ball's approach to cube slicing, carried out for functions. Motivated by
inequalities for dual affine quermassintegrals, we also prove an isoperimetric
inequality for certain averages of the marginals of such $f$ for which the cube
is the extremal case.
