Random Steiner symmetrizations of sets and functions
Random Steiner symmetrizations of sets and functions
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集合和函数的随机 Steiner 对称化
DOI：
10.1007/s00526-012-0493-4
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发表时间：
2013
影响因子：
2.1
通讯作者：
A. Volčič
中科院分区：
文献类型：
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作者：
A. Volčič
本文证明了具有有限测度的可测集合的随机斯坦纳对称序列与具有相同测度的球在L1距离上的几乎肯定收敛性。从这一结果，我们推导出非负Lp函数在自然范数下随机Steiner对称的球对称几乎肯定收敛和有界支持下非负连续函数一致收敛的类似命题。后一个结果最后被用来证明紧集合的随机对称序列在到具有相同测度的球的Hausdorff距离上几乎肯定收敛，提供了Van Schaftingen在2006年给出的另一个证明Mani-Levitska猜想（Topol Methods Nonlinear Anal 28(1): 61-85, 2006）。
In this article we prove almost sure convergence, in the L1 distance, of sequences of random Steiner symmetrizations of measurable sets having finite measure to the ball having the same measure. From this result we deduce analogous statements concerning the almost sure convergence to the spherical symmetrization of random Steiner symmetrizations of non negative Lp functions in the natural norm and uniform convergence of non negative continuous functions with bounded support. The latter result is finally used to prove that sequences of random symmetrizations of a compact set converge almost surely in the Hausdorff distance to the ball having the same measure, providing another proof of Mani-Levitska’s conjecture besides the one given in 2006 by Van Schaftingen (Topol Methods Nonlinear Anal 28(1): 61–85, 2006).