论文标题
凸面的局部结构附近和圆锥形点附近
Local structure of convex surfaces near regular and conical points
论文作者
论文摘要
考虑$ \ mathbb {r}^d $，$ d \ ge 2 $的凸表面上的一个点，此时向表面上的支撑$π$平面。绘制平行于$π$切割一部分表面的平面。当飞机接近点时，我们研究表面的这一部分的限制行为，总是与$π$平行。更确切地说，我们研究了表面的这一部分引起的$ s^{d-1} $中标准化表面积测量的限制行为。在本文中，我们考虑两种情况：（a）当点是规则的，（b）当它是奇异的圆锥形时，即，在该点处的切线不包含直线。在情况下（a），极限是位于外向正常矢量的原子至$π$，在（b）的情况下，极限等于由平面切断的切线锥部分引起的度量。
Consider a point on a convex surface in $\mathbb{R}^d$, $d \ge 2$ and a plane of support $Π$ to the surface at this point. Draw a plane parallel to $Π$ 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 $Π$. 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 $Π$, 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.
