Local structure of convex surfaces near regular and conical points

TL;DR

This paper investigates the local geometric behavior of convex surfaces near regular and conical points by analyzing the limiting surface area measures as a supporting plane approaches the point.

Contribution

It provides a detailed description of the limiting behavior of surface area measures at convex surface points, distinguishing between regular and conical singularities.

Findings

At regular points, the limit measure is an atom at the outward normal vector.

At conical singular points, the limit measure is induced by the tangent cone.

The results clarify the local structure of convex surfaces near different types of 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.