关于平行集的体积和表面积
On Volume and Surface Area of Parallel Sets
摘要：
在欧几里得空间中，集合A的r-平行集由所有与A的距离不超过r的点组成。我们澄清的体积和平行集的表面积之间的关系，并研究这两个数量的渐近行为，r趋于0。例如，我们表明，在一般情况下，存在（适当重新缩放）限制的表面积意味着存在相应的限制的体积，称为闵可夫斯基内容。自相似分形集的情况下，得到一个完整的特征。平稳随机集的应用进行了讨论，以及，特别是对轨迹的布朗运动。
英文摘要：
The r-parallel set to a set A in a Euclidean space consists of all points with distance at most r from A. We clarify the relation between the volume and the surface area of parallel sets and study the asymptotic behaviour of both quantities as r tends to 0. We show, for instance, that in general, the existence of a (suitably rescaled) limit of the surface area implies the existence of the corresponding limit for the volume, known as the Minkowski content. A full characterisation is obtained for the case of self-similar fractal sets. Applications to stationary random sets are discussed as well, in particular, to the trajectory of the Brownian motion.
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