Average decay of Fourier transforms and geometry of convex sets

Abstract

Let $B$ be a convex body in ${\mathbb{R}}^2$ with piecewise smooth boundary and let ${\hat{\chi }}_B$ denote the Fourier transform of its characteristic function. In this paper we determine the admissible decays of the spherical $L^p$-averages of ${\hat{\chi }}_B$ 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.