Fourier transform, null variety, and Laplacian's eigenvalues

摘要

We consider a quantity kappa(Omega)-the distance to the origin from the null variety of the Fourier transform of the characteristic function of Omega. We conjecture, firstly, that kappa(Omega) is maximised, among all convex balanced domains Omega subset of R-d of a fixed volume, by a ball, and also that kappa(Omega) is bounded above by the square root of the second Dirichlet eigenvalue of Omega. We prove some weaker versions of these conjectures in dimension two, as well as their validity for domains asymptotically close to a disk, and also discuss further links between kappa(Omega) and the eigenvalues of the Laplacians.