Local structure of convex surfaces near regular and conical points

Abstract

Consider a point on a convex surface in $\mathbb{R}^d$, $d \ge 2$ and a plane of support $\Pi$ to the surface at this point. Draw a plane parallel to $\Pi$ cutting a part of the surface. We study the limiting behavior of this part of surface when the plane approaches the point, being always parallel to $\Pi$. More precisely, we study the limiting behavior of the normalized surface area measure in $S^{d-1}$ induced by this part of 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 the case (a) the limit is the atom located at the outward normal vector to $\Pi$, and in the case (b) the limit is equal to the measure induced by the part of the tangent cone cut off by a plane.