We prove that the Teichmüller space of surfaces of genus $g$ with $p$ punctures contains balls which are not convex in the Teichmüller metric whenever its complex dimension $(3g −3+p)$ is greater than $1$.
"Non-convex balls in the Teichmüller metric." J. Differential Geom. 110 (3) 379 - 412, November 2018. https://doi.org/10.4310/jdg/1542423625