On sharp bounds for marginal densities of product measures

الأصل · EN

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 ⁿ of the form f(x)=∏ᵢ₌₁ⁿ fᵢ(xᵢ), where each fᵢ is a density on, 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.