Local structure of convex surfaces near regular and conical points

Abstract

Consider a point on a convex surface in R^d, d ≥ 2 and a plane of support Π to the surface at this point. Draw a plane parallel with Π cutting a part of the surface. We study the limiting behavior of this part of the surface when the plane approaches the point, being always parallel with Π. More precisely, we study the limiting behavior of the normalized surface area measure in S^{d−1} induced by this part of the surface. In this paper, we consider two cases: (a) when the point is regular and (b) when it is singular conical, that is the tangent cone at the point does not contain straight lines. In Case (a), the limit is the atom located at the outward normal vector to Π, and in Case (b), the limit is equal to the measure induced by the part of the tangent cone cut off by a plane.