Besicovitch Covering Property for homogeneous distances in the Heisenberg groups
Besicovitch Covering Property for homogeneous distances in the Heisenberg groups
复制标题
海森堡群中均匀距离的贝西科维奇覆盖性质
DOI：
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
Séverine Rigot
中科院分区：
文献类型：
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作者：
E. Donne;Séverine Rigot
我们的主要结果是一个肯定的答案的问题是否可以找到齐次距离的海森堡群，有贝西科维奇覆盖性质（BCP）。这个性质是众所周知的测度论的基本工具之一，与测度微分理论有着密切的联系。证明了以原点为中心的单位球与欧氏球重合的齐次距离满足BCP。这样的齐次距离确实存在于任何卡诺群的结果Hebisch和Sikora。在海森堡群中，它们与Cygan-Koranyi（也称为Koranyi）距离有关。他们被认为特别是李和Naor提供了一个反例的戈曼-线性猜想在理论计算机科学。为了使我们的结果的角度来看，我们还证明了两个几何标准，暗示BCP的无效性，表明在某种意义上我们的例子是尖锐的。我们的第一个标准特别适用于海森堡群上常用的齐次距离，例如Cygan-Koranyi距离和Carnot-Caratheodory距离，这些距离已知不满足BCP。为了从不同的角度来看待这些结果，并为了完整性，我们也给出了D. Preiss的理论，在一般度量空间中，人们总是可以构造一个不满足BCP的双Lipschitz等价距离。
Our main result is a positive answer to the question whether one can find homogeneous distances on the Heisenberg groups that have the Besicovitch Covering Property (BCP). This property is well known to be one of the fundamental tools of measure theory, with strong connections with the theory of differentiation of measures. We prove that BCP is satisfied by the homogeneous distances whose unit ball centered at the origin coincides with an Euclidean ball. Such homogeneous distances do exist on any Carnot group by a result of Hebisch and Sikora. In the Heisenberg groups, they are related to the Cygan-Koranyi (also called Koranyi) distance. They were considered in particular by Lee and Naor to provide a counterexample to the Goemans-Linial conjecture in theoretical computer science. To put our result in perspective, we also prove two geometric criteria that imply the non-validity of BCP, showing that in some sense our example is sharp. Our first criterion applies in particular to commonly used homogeneous distances on the Heisenberg groups, such as the Cygan-Koranyi and Carnot-Caratheodory distances that are already known not to satisfy BCP. To put a different perspective on these results and for sake of completeness, we also give a proof of the fact, noticed by D. Preiss, that in a general metric space, one can always construct a bi-Lipschitz equivalent distance that does not satisfy BCP.