Gabor orthogonal bases and convexity

Abstract: Let $g(x)=\chi_B(x)$ be the indicator function of a bounded convex set in $\Bbb R^d$, $d\geq 2$, with a smooth boundary and everywhere non-vanishing Gaussian curvature. Using a combinatorial appraoch we prove that if $d \neq 1 \mod 4$, then there does not exist $S \subset {\Bbb R}^{2d}$ such that ${ {g(x-a)e^{2 \pi i x \cdot b} }}_{(a,b) \in S}$ is an orthonormal basis for $L^2({\Bbb R}^d)$.