  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 maximized, among all
convex balanced domains $\Omega\subset\Rbb^d$ 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.
