On a problem of Turán about positive definite functions

We study the following question posed by Turán. Suppose $\Omega$ is a convex body in Euclidean space $\mathbb{R} ^d$ which is symmetric in $\Omega$ and with value $1$ at the origin; which one has the largest integral? It is probably the case that the extremal function is the indicator of the half-body convolved with itself and properly scaled, but this has been proved only for a small class of domains so far. We add to this class of known Turán domains the class of all spectral convex domains. These are all convex domains which have an orthogonal basis of exponentials $e_\lambda(x) = \exp 2\pi i\langle{\lambda}{x}\rangle$, $\lambda \in \mathbb{R} ^d$. As a corollary we obtain that all convex domains which tile space by translation are Turán domains.

We also give a new proof that the Euclidean ball is a Turán domain.