关于度量三元组的曲率
On the Curvature of Metric Triples
摘要：
在这篇文章中，我们引入了曲率的概念，表示为$ k_X(T)$，度量三元组$T$内（可能离散）度量空间$X$。这样的概念使我们能够考虑任何度量空间的曲率信息，包括离散的度量空间，如由科学数据产生的那些。为了定义这一概念，我们采用了由三元组的边长以及从三元组的顶点到度量空间的点的最小总距离组成的信息。这些信息为我们提供了一个唯一的数字$k_X(T)$，使得三重$T$可以等距嵌入到模型空间$M_k^2$中，直到$k\le k_X(T)$。当$X$是模型空间的凸子集时，值$k_X(T)$与通常的曲率一致。我们还证明了在$CAT(k)$空间内的任何度量三元组$T$的曲率$k_X(T)$上有界于$k$。
英文摘要：
In this article, we introduce a notion of curvature, denoted by $ k_X(T)$, for a metric triple $T$ inside a (possibly discrete) metric space $X$. Such a notion enables us to consider curvature information of any metric space, including discrete metric spaces such as those generated by scientific data. To define the notion, we employ the information consisting of side lengths of the triple as well as the minimum total distance from vertices of the triple to points of the metric space. Those information provides us a unique number $k_X(T)$ such that the triple $T$ can be isometrically embedded into the model space $M_k^2$ up to $k\le k_X(T)$. The value $k_X(T)$ agrees with the usual curvature when $X$ is a convex subset of a model space. We also show that the curvature $k_X(T)$ of any metric triple $T$ inside a $CAT(k)$ space is bounded above by $k$.
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