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

Abstract

We prove that no smooth symmetric convex body Ω with at least one point of non-vanishing Gaussian curvature can admit an orthogonal basis of exponentials. (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 Ω can tile [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] d by translations.