On the Curvature of Metric Triples

Abstract: 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 $k$.