Random projections of regular polytopes (Q1969093)

Based on an approach of \textit{F. Affentranger} and \textit{R. Schneider} [Discrete Comput. Geom. 7, No. 3, 219-226 (1992; Zbl 0751.52002)], the authors give an asymptotic formula for the expected number of \(k\)-faces of the orthogonal projection of a regular \(n\)-crosspolytope in \(\mathbb{R}^n\) onto a random \(d\)-dimensional isotropic subspace, as \(n\) tends to infinity. The authors also present an asymptotic formula (with all constants given) for the (spherical) volume of spherical regular simplices, which generalizes Daniel's formula. Note that the expected number of \(k\)-faces in the projection coincides with the expected number of \(k\)-faces of the convex hull of a standard Gaussian sample of \(n\) pairs \(x^1,-x^1,\dots, x^n,-x^n\) of points in \(\mathbb{R}^d\).