  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.
