Mean width of regular polytopes and expected maxima of correlated Gaussian variables

An old conjecture states that among all simplices inscribed in the unit sphere, the regular one has the maximal mean width. We restate this conjecture probabilistically and prove its asymptotic version. We also show that the mean width of the regular simplex with $2n$ vertices is remarkably close to the mean width of the regular crosspolytope with the same number of vertices. We establish several formulas conjectured by S. Finch on projection length $W$ of the regular cube, simplex and crosspolytope onto a line with random direction. Finally, we prove distributional limit theorems for $W$ as the dimension of the regular polytope goes to $\infty$.