In the paper
Yau, Shing-Tung, Non-existence of continuous convex functions on certain Riemannian manifolds, Math. Ann. 207, 269-270 (1974). ZBL0261.53036., S.T.Yau proved that there does not exists a non-trivial convex function on complete Riemannian manifold with finite volume. But the function $f=|x|$ is convex on $[-1,1]$ and $[-1,1]$ has finite volume. I am confused about that. Please help me. Thank you