Geometric bounds for Favard length
Geometric bounds for Favard length
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Favard 长度的几何界限
DOI：
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发表时间：
2017
影响因子：
1
通讯作者：
Tyler Bongers
中科院分区：
文献类型：
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作者：
Tyler Bongers
给定平面上的一个集合，它在所有方向上的投影的平均长度称为法瓦德长度。这个量测量集合的大小，并且与集合的度量和几何性质密切相关，例如可求长性，Hausdorff维数和分析能力。在本文中，我们开发了新的几何技术估计Favard长度。我们将给出一个简短的几何动机的证据有关豪斯多夫维数的衰减率的Favard长度的邻域的一组。我们还将表明，序列的Favard长度的世代的自相似集是凸的，这有直接的应用程序，给下界Favard长度的各种分形集。
Given a set in the plane, the average length of its projections over all directions is called Favard length. This quantity measures the size of a set, and is closely related to metric and geometric properties of the set such as rectifiability, Hausdorff dimension, and analytic capacity. In this paper, we develop new geometric techniques for estimating Favard length. We will give a short geometrically motivated proof relating Hausdorff dimension to the decay rate of the Favard length of neighborhoods of a set. We will also show that the sequence of Favard lengths of the generations of a self-similar set is convex; this has direct applications to giving lower bounds on Favard length for various fractal sets.