We use the convexity of a certain function discovered by W. Kendall on small metric balls in CAT(1)-spaces to show that any probability measure on a complete CAT(1)-space of small radius admits a unique barycenter. We also present various properties of barycenter on those spaces. This extends the results previously known for CAT(0)-spaces and CAT(1)-spaces of small diameter.
Takumi YOKOTA. "Convex functions and barycenter on CAT(1)-spaces of small radii." J. Math. Soc. Japan 68 (3) 1297 - 1323, July, 2016. https://doi.org/10.2969/jmsj/06831297