Moderate smoothness of most Alexandrov surfaces

We show that, in the sense of Baire category, most Alexandrov surfaces with curvature bounded below by $kappa$ have no conical points. We use this result to prove that at most points of such surfaces, the lower and the upper Gaussian curvatures are equal to $kappa$ and $infty$ respectively.

On the Theorem of the Three Perpendiculars

We show that the theorem of the three perpendiculars holds in any n-dimensional space form.