Differentiating maps into L1, and the geometry of BV functions
Differentiating maps into L1, and the geometry of BV functions
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将映射微分为 L1 和 BV 函数的几何
DOI：
10.4007/annals.2010.171.1347
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发表时间：
2006
影响因子：
4.9
通讯作者：
B. Kleiner
中科院分区：
文献类型：
--
作者：
J. Cheeger;B. Kleiner
这是一系列论文之一，研究Lipschitz映射X → V的微分理论和双Lipschitz不可嵌入性之间的相互作用，其中X是度量测度空间，V是Banach空间。在这里，我们考虑的情况下V = L 1，可微性失败。我们建立了另一种可微性的某些X，包括Ewn和H，海森堡群与它的Carnot-Caratheodory度量。因此，L1中的双Lipschitz嵌入不是如J. Lee和A. Naor.当与他们的工作相结合时，这为理论计算机科学中的Goemans-Linial猜想提供了一个自然的反例;第一个这样的反例由Khot-Vishnoi发现[KV 05]。证明我们的主要定理的一个关键因素是Lipschitz映射到L 1和有界变差函数之间的新联系，这使我们能够利用海森堡群上BV函数结构的结果[FSSC 01]。
This is one of a series of papers examining the interplay between differentiation theory for Lipschitz maps X → V and bi-Lipschitz nonembeddability, where X is a metric measure space and V is a Banach space. Here, we consider the case V = L 1 , where differentiability fails. We establish another kind of differentiability for certain X, including ℝ n and H, the Heisenberg group with its Carnot-Caratheodory metric. It follows that ℍ does not bi-Lipschitz embed into L 1 , as conjectured by J. Lee and A. Naor. When combined with their work, this provides a natural counterexample to the Goemans-Linial conjecture in theoretical computer science; the first such counterexample was found by Khot-Vishnoi [KV05]. A key ingredient in the proof of our main theorem is a new connection between Lipschitz maps to L 1 and functions of bounded variation, which permits us to exploit results on the structure of BV functions on the Heisenberg group [FSSC01].