  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
exponentials. (The non-symmetric case was proven by 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.
