Let µ be an n-dimensional probability measure with a bounded density f. We prove that most marginals of f are well-bounded. We show that this probabilistic fact is based on affine invariance properties and extremal inequalities for certain averages of f on the Grassmannian and affine Grassmannian. These inequalities can be viewed as functional analogues of affine isoperimetric inequalities for convex sets due to Busemann–Straus, Grinberg and Schneider. This talk will be based on a joint work with Susanna Dann and Peter Pivovarov.