正定度量空间
Positive definite metric spaces
摘要：
模是有限度量空间的一个数值不变量，最近由T. Leinster，它在精确意义上类似于有限集的基数或拓扑空间的欧拉示性数。它已被推广到无限度量空间在几个先验的不同方式。本文发展了一类度量空间的理论，即正定度量空间，它的大小比一般的度量空间更容易处理。正定性是度量空间的经典负型性质的推广，已知它对许多有趣的空间类都成立。本文证明了所有提出的度量定义在紧正定度量空间上都是一致的，并进一步证明了度量作为这类空间的函数的性质。最后，证明了p \le 2的l_p^n的紧子集的大小，推广了Leinster关于p= 1，2的结果，并利用了这些空间的比正定性强一些的性质.
英文摘要：
Magnitude is a numerical invariant of finite metric spaces, recently introduced by T. Leinster, which is analogous in precise senses to the cardinality of finite sets or the Euler characteristic of topological spaces. It has been extended to infinite metric spaces in several a priori distinct ways. This paper develops the theory of a class of metric spaces, positive definite metric spaces, for which magnitude is more tractable than in general. Positive definiteness is a generalization of the classical property of negative type for a metric space, which is known to hold for many interesting classes of spaces. It is proved that all the proposed definitions of magnitude coincide for compact positive definite metric spaces and further results are proved about the behavior of magnitude as a function of such spaces. Finally, some facts about the magnitude of compact subsets of l_p^n for p \le 2 are proved, generalizing results of Leinster for p=1,2, using properties of these spaces which are somewhat stronger than positive definiteness.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
