The deviation of a general convex body with twice differentiable boundary and an arbitrarily positioned polytope with a given number of vertices is studied. The paper considers the case where the deviation is measured in terms of the surface areas of the involved sets, more precisely, by what is called the surface area deviation. The proof uses arguments and constructions from probability, convex and integral geometry. The bound is closely related to p-affine surface areas.

中文翻译：

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光滑凸体和多面体之间的表面积偏差

研究了具有两个可微边界的一般凸体和具有给定数量顶点的任意定位的多边形的偏差。本文考虑了这样一种情况，即根据所涉及集合的表面积来衡量偏差，更确切地说，是通过所谓的表面积偏差来衡量的。该证明使用了概率，凸几何和积分几何的论点和构造。该界与p-仿射表面积密切相关。