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

We prove that no smooth symmetric convex body Omega with at least one point of non-vanishing Gaussian curvature can admit an orthogonal basis of expon entials. (The nonsymmetric case was proven in a preprint by M. Kolountzakis ). This is further evidence of Fuglede's conjecture, which states that such a basis is possible if and only if Omega can tile R-d by translations.