  The intersections of beta-Voronoi, beta-prime-Voronoi and Gaussian-Voronoi
tessellations in $\mathbb{R}^d$ with $\ell$-dimensional affine subspaces,
$1\leq \ell\leq d-1$, are shown to be random tessellations of the same type but
with different model parameters. In particular, the intersection of a classical
Poisson-Voronoi tessellation with an affine subspace is shown to have the same
distribution as a certain beta-Voronoi tessellation. The geometric properties
of the typical cell and, more generally, typical $k$-faces, of the sectional
Poisson-Voronoi tessellation are studied in detail. It is proved that in high
dimensions, that is as $d\to\infty$, the intersection of the $d$-dimensional
Poison-Voronoi tessellation with an affine subspace of fixed dimension $\ell$
converges to the $\ell$-dimensional Gaussian-Voronoi tessellation.
