## On a problem of Turán about positive definite functions

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- by Mihail N. Kolountzakis and Szilárd Gy. Révész
- Proc. Amer. Math. Soc.
**131**(2003), 3423-3430 - DOI: https://doi.org/10.1090/S0002-9939-03-07023-0
- Published electronically: March 25, 2003
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## Abstract:

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.

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## Bibliographic Information

**Mihail N. Kolountzakis**- Affiliation: Department of Mathematics, University of Crete, Knossos Ave., 714 09 Iraklio, Greece
- Email: kolount@member.ams.org
**Szilárd Gy. Révész**- Affiliation: Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, 1364 Budapest, Hungary
- Email: revesz@renyi.hu
- Received by editor(s): May 22, 2002
- Published electronically: March 25, 2003
- Additional Notes: The first author was supported in part by INTAS grant, project 99-01080

The second author was supported in part by the Hungarian National Foundation for Scientific Research, Grant # T034531 and T032872. - Communicated by: Andreas Seeger
- © Copyright 2003 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**131**(2003), 3423-3430 - MSC (2000): Primary 42B10; Secondary 26D15, 52C22, 42A82, 42A05
- DOI: https://doi.org/10.1090/S0002-9939-03-07023-0
- MathSciNet review: 1990631