Average decay of Fourier transforms and geometry of convex sets (Q1283341)

Given a convex body \(B\), that is a compact convex set with nonempty interior in \(\mathbb{R}^n\), denote by \(\chi_B\) its characteristic function. The authors study the decay of the Fourier transform \[ \widehat\chi_B(\xi):= \int_B e^{-2\pi i\xi\cdot x} dx\quad\text{as }|\xi|\to \infty, \] in terms of the geometric properties of \(B\). The admissible decays of the spherical \(L^p\)-averages \[ \Biggl(\int_{\Sigma_{n-1}} |\widehat\chi_B(\rho\sigma)|^p d\sigma\Biggr)^{1/p},\quad 1< p\leq\infty, \] are determined, where \(\Sigma_{n-1}\) is the unit sphere in \(\mathbb{R}^n\) and \(d\sigma\) is the Lebesgue measure on it. As an application, sharp results are obtained on the average number of integer lattice points in ``large'' convex planar bodies.