On the Curvature of Metric Triples

TL;DR

This paper introduces a new curvature measure for metric triples within metric spaces, including discrete data, which generalizes classical curvature and is bounded in CAT(k) spaces.

Contribution

It defines a novel curvature notion for metric triples that applies to any metric space, extending classical concepts to discrete and complex spaces.

Findings

The curvature $k_X(T)$ can be uniquely determined from side lengths and distances.

It coincides with classical curvature in convex subsets of model spaces.

In CAT(k) spaces, the curvature of triples is bounded above by $k$.

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 $k$.