集合的高阶点态可微性
Pointwise differentiability of higher order for sets
摘要：
本文通过比较欧几里得空间中任意子集的距离函数与光滑子流形的距离函数，给出了该空间中任意子集的高阶点态可微性的两个概念。结果包括微分是Borel函数，高阶可求微点集的可求性，以及Rademacher结果。一个概念的特点是由一个极限程序涉及非均匀扩张集。最初的动机制定的概念源于研究的支持平稳积分varifolds。特别地，在支撑与给定平面相交的几乎所有点处，证明了每个正整数阶的强逐点可微性。
英文摘要：
The present paper develops two concepts of pointwise differentiability of higher order for arbitrary subsets of Euclidean space defined by comparing their distance functions to those of smooth submanifolds. Results include that differentials are Borel functions, higher order rectifiability of the set of differentiability points, and a Rademacher result. One concept is characterised by a limit procedure involving inhomogeneously dilated sets. The original motivation to formulate the concepts stems from studying the support of stationary integral varifolds. In particular, strong pointwise differentiability of every positive integer order is shown at almost all points of the intersection of the support with a given plane.
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