Galyna V. Livshyts

On sharp bounds for marginal densities of product measures

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 Rn of the form f(x) = Пi=1nfi(xi), where each fi is a density on R, say bounded by one, then the density of any marginal πE(f) is bounded by 2k/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.