On a Problem of Turán about Positive Definite Functions

We study the following question posed by Turán. Suppose Ω is a convex body in Euclidean space Rd which is symmetric in Ω 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 eλ(x) = exp 2πi〈λx〉, λ ∈ Rd. 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. 0. A problem of Turán The following question is attributed to Turán (and Stechkin [1]): (Turán) Let Ω be a convex domain in R which is symmetric with respect to 0. What is the maximum of ∫ f for all f supported in Ω which are positive definite (their Fourier Transform is nonnegative) and have f(0) = 1? See [2, 5, 13] for the history of the problem. Definition 1 (Turán domains). A symmetric convex domain Ω is called a Turán domain if the maximum of ∫ f over all positive definite functions supported in Ω and with f(0) = 1 is achieved by the function (1) f = 2 |Ω| 2 Ω ∗ χ 2 Ω. Otherwise it is called an anti-Turán domain. Received by the editors May 22, 2002. 2000 Mathematics Subject Classification. Primary 42B10; Secondary 26D15, 52C22, 42A82, 42A05.