Average decay of Fourier transforms and geometry of convex sets

Let B be a convex body in R-2, with piecewise smooth boundary and let <(chi)over cap>(B) denote the Fourier transform of its characteristic function. In this paper we determine the admissible decays of the spherical L-p-averages of <(chi)over cap>(B) and we relate our analysis to a problem in the geometry of convex sets. As an application we obtain sharp results on the average number of integer lattice points in large bodies randomly positioned in the plane