Random projections of regular polytopes

Based on an approach of Affentranger & Schneider we give an asymptotic form ula for the expected number of k-faces of the orthogonal projection of a re gular n-crosspolytope onto a randomly chosen isotropic subspace of fixed di mension, as n tends to infinity. In particular, we present a precise asympt otic formula for the (spherical) volume of spherical regular simplices, whi ch generalizes Daniel's formula.