Mohammed Ghomi's answer reminds me of a related picture that Cedric Villani drew to depict Ricci curvature ([1] Chapter 14). Similar to the $CAT(\kappa)$ inequality, this idea can be used to derive notions of Ricci curvature for more general metric measure spaces.
[![enter image description here][2]][2]


  [2]: https://i.sstatic.net/rdgDr.png

[1] <cite authors="Villani, Cédric">_Villani, Cédric_, [**Optimal transport. Old and new**](http://dx.doi.org/10.1007/978-3-540-71050-9), Grundlehren der Mathematischen Wissenschaften 338. Berlin: Springer (ISBN 978-3-540-71049-3/hbk). xxii, 973&nbsp;p. (2009). [ZBL1156.53003](https://zbmath.org/?q=an:1156.53003).</cite>