Universal points of convex bodies and bisectors in Minkowski spaces

We deal with different properties of a smooth and strictly convex body that depend on the behavior of the planar sections of the body parallel to and close to a given tangent plane. The first topic is boundary points where any given convex domain in the tangent plane can be approximated by a sequence of suitably rescaled planar sections (so-called p-universal points). In the second topic, the given convex body is the unit ball of a Minkowski space, and oscillation properties of bisectors and trisectors in that space are considered. In either case it turns out that geometrically irregular behavior is typical, when considered from the viewpoint of Baire category.