  Let $P_n$ be an $n$-dimensional regular polytope from one of the three
infinite series (regular simplices, regular crosspolytopes, and cubes). Project
$P_n$ onto a random, uniformly distributed linear subspace of dimension $d\geq
2$. We prove that the expected number of $k$-dimensional faces of the resulting
random polytope is an increasing function of $n$. As a corollary, we show that
the expected number of $k$-faces of the Gaussian polytope is an increasing
function of the number of points used to generate the polytope. Similar results
are obtained for the symmetric Gaussian polytope and the Gaussian zonotope.
