  The notions of Hausdorff and Fourier dimensions are ubiquitous in harmonic
analysis and geometric measure theory. It is known that any hypersurface in
$\mathbb{R}^{d+1}$ has Hausdorff dimension $d$. However, the Fourier dimension
depends on the finer geometric properties of the hypersurface. For instance,
the Fourier dimension of a hyperplane is 0, and the Fourier dimension of a
hypersurface with non-vanishing Gaussian curvature is $d$. Recently, Harris has
shown that the Euclidean light cone in $\mathbb{R}^{d+1}$ has Fourier dimension
$d-1$, which leads one to conjecture that the Fourier dimension of a
hypersurface equals the number of non-vanishing principal curvatures. We prove
this conjecture for all $d$-dimensional cones and cylinders in
$\mathbb{R}^{d+1}$ generated by hypersurfaces in $\mathbb{R}^d$ with
non-vanishing Gaussian curvature. In particular, cones and cylinders are not
Salem. Our method involves substantial generalizations of Harris's strategy.
