On volume and surface area of parallel sets
On volume and surface area of parallel sets
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关于平行组的体积和表面积
DOI：
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发表时间：
2009
期刊：
影响因子：
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通讯作者：
S. Winter
中科院分区：
文献类型：
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作者：
J. Rataj;S. Winter
欧几里德空间中集合a的r-平行集合由距离a不超过r的所有点组成。我们阐明了平行集合的体积与表面积之间的关系，并研究了当r趋于0时这两个量的渐近行为。例如，我们证明，一般来说，表面积极限的存在（适当地重新标度）意味着相应体积极限的存在，即闵可夫斯基含量。得到了自相似分形集的一个完整刻画。本文还讨论了在平稳随机集上的应用，特别是在布朗运动轨迹上的应用。
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.