Average decay of Fourier transforms and geometry of convex sets

Let B be a convex body in R2, with piecewise smooth boundary and let ^?B denote the Fourier transform of its characteristic function. In this paper we determine the admissible decays of the spherical Lp averages of ^?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