  For two families of random polytopes we compute explicitly the expected sums
of the conic intrinsic volumes and the Grassmann angles at all faces of any
given dimension of the polytope under consideration. As special cases, we
compute the expected sums of internal and external angles at all faces of any
fixed dimension. The first family are the Gaussian polytopes defined as convex
hulls of i.i.d. samples from a non-degenerate Gaussian distribution in $\mathbb
R^d$. The second family are convex hulls of random walks with exchangeable
increments satisfying certain mild general position assumption. The expected
sums are expressed in terms of the angles of the regular simplices and the
Stirling numbers, respectively. There are non-trivial analogies between these
two settings. Further, we compute the angle sums for Gaussian projections of
arbitrary polyhedral sets, of which the Gaussian polytopes are a special case.
Also, we show that the expected Grassmann angle sums of a random polytope with
a rotationally invariant law are invariant under affine transformations. Of
independent interest may be also results on the faces of linear images of
polyhedral sets. These results are well known but it seems that no detailed
proofs can be found in the existing literature.
