度量空间的严格p-负型
Strict p-negative type of a metric space
摘要：
Doust和Weston提出了一种称为“增强负型”的新方法，用于计算任意给定的有限度量树（T，d）的上严格p-负型的非平凡下界p（T）。在有限度量树的上下文中，任何这样的下界p（T）> 1被认为是非平凡的。在本文中，我们改进了增强的负型技术，并展示了它如何可以更普遍地应用于任何有限度量空间（X，d），这是已知有严格的p-负型的一些非负的p.这使我们能够显着改善的下界上的有限度量树，由Doust和Weston给出，而且，导致我们的主要结果之一：有限度量空间的极大p-负型不可能是严格的。通过应用，我们可以证明对于某些p > 1，大类的有限度量空间（例如Hadamard流形的有限等距子空间）必须具有严格p-负型。我们还证明了如果一个度量空间（有限的或非有限的）对某些p > 0有p-负型，那么它对[0，p）中的所有q必有严格q-负型.这推广了Schoenberg的一个著名定理，并导出了度量空间可具有严格p-负型的区间的完全分类。（本文中的几个结果对于半度量空间更一般地成立。）
英文摘要：
Doust and Weston introduced a new method called "enhanced negative type" for calculating a non trivial lower bound p(T) on the supremal strict p-negative type of any given finite metric tree (T,d). In the context of finite metric trees any such lower bound p(T) > 1 is deemed to be non trivial. In this paper we refine the technique of enhanced negative type and show how it may be applied more generally to any finite metric space (X,d) that is known to have strict p-negative type for some non negative p. This allows us to significantly improve the lower bounds on the supremal strict p-negative type of finite metric trees that were given by Doust and Weston and, moreover, leads in to one of our main results: The supremal p-negative type of a finite metric space cannot be strict. By way of application we are then able to exhibit large classes of finite metric spaces (such as finite isometric subspaces of Hadamard manifolds) that must have strict p-negative type for some p > 1. We also show that if a metric space (finite or otherwise) has p-negative type for some p > 0, then it must have strict q-negative type for all q in [0,p). This generalizes a well known theorem of Schoenberg and leads to a complete classification of the intervals on which a metric space may have strict p-negative type. (Several of the results in this paper hold more generally for semi-metric spaces.)
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