Angle Sums of Random Polytopes

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