Local structure of convex surfaces

Abstract

A point on the surface of a convex body and a supporting plane to the body at this point are under consideration. A plane parallel to this supporting plane and cutting off part of the surface is drawn. The limiting behaviour of the cut-off part of the surface as the cutting plane approaches the point in question is investigated. More precisely, the limiting behavior of the appropriately normalized surface area measure in $S^2$ generated by this part of the surface is studied. The cases when the point is regular and singular (a conical or a ridge point) are considered. The supporting plane can be positioned in different ways with respect to the tangent cone at the point: its intersection with the cone can be a vertex, a line (if a ridge point is considered), a plane angle (which can degenerate into a ray or a half-plane), or a plane (if the point is regular and, correspondingly, the cone degenerates into a half-space). In the case when the intersection is a ray, the plane can be tangent (in a one- or two-sided manner) or not tangent to the cone. It turns out that the limiting behaviour of the measure can be different. In the case when the intersection of the supporting plane and the cone is a vertex or in the case of a (one- or two-sided) tangency, the weak limit always exists and is uniquely determined by the plane and the cone. In the case when the intersection is a line or a ray with no tangency, there may be no limit at all. In this case all possible weak partial limits are characterized. Bibliography: 13 titles.