Let µ be a probability measure on 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 µ, any marginal π E (µ), or a small perturbation thereof, satisfies a nearly optimal small-ball probability.

A flag area measure on an n-dimensional euclidean vector space is a continuous translation-invariant valuation with values in the space of signed measures on the flag manifold consisting of a unit vector v and a (p+1)-dimensional linear subspace containing v with 0 ≤ p ≤ n−1.Using local parallel sets, Hinderer constructed examples of SO(n)covariant flag area measures. There is an explicit formula for his flag area measures evaluated on polytopes, which involves the squared cosine of the angle between two subspaces.We construct a more general sequence of smooth SO(n)-covariant flag area measures via integration over the normal cycle of appropriate differential forms. We provide an explicit description of our measures on polytopes, which involves an arbitrary elementary symmetric polynomial in the squared cosines of the principal angles between two subspaces.Moreover, we show that these flag area measures span the space of all smooth SO(n)-covariant flag area measures, which gives a classification result in the spirit of Hadwiger's theorem.2010 Mathematics Subject Classification. Primary 52A39, Secondary 52B45.

Abstract. The Busemann-Petty problem asks whether origin-symmetric convex bodies in R n with smaller central hyperplane sections necessarily have smaller volume. The answer is affirmative if n ≤ 4 and negative if n ≥ 5. We study this problem in the complex hyperbolic n-space H n C and prove that the answer is affirmative for n ≤ 2 and negative for n ≥ 3.

Building on work of Furstenberg and Tzkoni, we introduce r-flag affine quermassintegrals and their dual versions. These quantities generalize affine and dual affine quermassintegrals as averages on flag manifolds (where the Grassmannian can be considered as a special case). We establish affine and linear invariance properties and extend fundamental results to this new setting. In particular, we prove several affine isoperimetric inequalities from convex geometry and their approximate reverse forms. We also introduce functional forms of these quantities and establish corresponding inequalities.

Busemann's intersection inequality asserts that the only maximizers of the integral S n−1 |K ∩ ξ ⊥ | n dξ among all convex bodies of a fixed volume in R n are centered ellipsoids. We study this question in the hyperbolic and spherical spaces, as well as general measure spaces.