  We study the following question posed by Turan. Suppose K is a convex body in
Euclidean space which is symmetric with respect to the origin. Of all positive
definite functions supported in K, 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 "Turan domains" the class of all spectral convex domains. These
are all convex domains which have an orthogonal basis of complex exponentials.
As a corollary we obtain that all convex domains which tile space by
translation are Turan domains. We also give a new proof that the Euclidean ball
is a Turan domain.
