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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. Forproduct measures, studied recently by Rudelson and Vershynin, our results showthere is a trade-off between the strength of such bounds and the probabilitywith which they hold. Our proof rests on new affinely-invariant extremalinequalities for certain averages of $f$ on the Grassmannian and affineGrassmannian. These are motivated by Lutwak's dual affine quermassintegrals forconvex sets. We show that key invariance properties of the latter, due toGrinberg, extend to families of functions. The inequalities we obtain can beviewed as functional analogues of results due to Busemann--Straus, Grinberg andSchneider. As an application, we show that without any additional assumptionson $\mu$, any marginal $\pi_E(\mu)$, or a small perturbation thereof, satisfiesa nearly optimal small-ball probability.

Bounding marginal densities via affine isoperimetry