有界Apollonian嵌入的一致压缩原理
A uniform contraction principle for bounded Apollonian embeddings
摘要：
设H表示实希尔伯特空间的标准单点完备。给定H的任何非平凡真子集U，可以定义U上的所谓“阿波罗”度量d_U。当U \subset V \subset H是嵌套真子集时，我们证明了它们的阿波罗度量满足一致收缩原理：设D=diam_V（U）\in [0，+ \infty ]是小子集相对于大子集的直径.则对于U中的每个x，y，我们有d_V（x，y）\leq tanh（D/4）d_U（x，y）。在一维空间中，伯克霍夫为RP ^[1]上有限段的希尔伯特度量建立了这个收缩原理。在第二个维度，它是由杜波依斯为子集的黎曼领域。这里所述的一般性是新的。
英文摘要：
Let H denote the standard one-point completion of a real Hilbert space. Given any non-trivial proper sub-set U of H one may define the so-called `Apollonian' metric d_U on U. When U \subset V \subset H are nested proper subsets we show that their associated Apollonian metrics satisfy the following uniform contraction principle: Let D=diam_V (U) \in [0,+\infty] be the diameter of the smaller subsets with respect to the large. Then for every x,y in U we have d_V(x,y) \leq tanh (D/4) d_U(x,y). In dimension one, this contraction principle was established by Birkhoff for the Hilbert metric of finite segments on RP^1. In dimension two it was shown by Dubois for subsets of the Riemann sphere. It is new in the generality stated here.
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