Large Curvature on Typical Convex Surfaces

We show in this paper that on most convex surfaces there exist points with arbitrarily large lower curvature in every tangent direction. Moreover, we show that, astonishingly, on most convex surfaces, although the set of points with curvature 0 in every tangent direction has full measure, it contains no pair of opposite points, i.e. points admitting parallel supporting planes