Convex bodies with a point of curvature do not have Fourier bases (Q2731703)

The authors study the Fuglede conjecture which states that a domain \(\Omega\subset {\mathbb R}^d\) admits a spectrum if and only if it tiles \({\mathbb R}^d\) by a family of translates of \(\Omega\). Originally, Fuglede proved this conjecture under the additional assumption that the tiling set or the spectrum are lattice subsets of \({\mathbb R}^d\). It has recently been studied by \textit{P. E. T. Jorgensen} and \textit{S. Pedersen} [Electron. Res. Announc. Am. Math. Soc. 4, No. 6, 35-42 (1998; Zbl 0893.28005)] and \textit{J. Lagarias} and \textit{Y. Wang} [J. Funct. Anal. 145, 73-98 (1997; Zbl 0898.47002)]. \textit{M. N. Kolountzakis} [Ill. J. Math. 44, No. 3, 542-550 (2000; Zbl 0972.52011)] proved that any nonsymmetric convex body \(\Omega\subset {\mathbb R}^d\) with at least one point of nonvanishing Gaussian curvature does not admit a spectrum. It turns out there that the symmetric thing appears an obstacle to these methods. In this paper, the authors prove in an extremely simple way that no smooth (or, piecewise smooth) symmetric convex domain \(\Omega\subset {\mathbb R}^d\) with at least one point of nonvanishing Gaussian curvature can admit a spectrum.