Picture below is from 160 page of Hamilton, Richard S., Four-manifolds with positive curvature operator, J. Differ. Geom. 24, 153-179 (1986). ZBL0628.53042.
I understand $f$ as a map from $M$ to $R^k$ according to the paper. And $X$ is a closed convex subset of $R^k$. How to understand the $f$ can be saw as a point of $\partial X$ ?
Besides, how a point subset $U\subset R^k$ containing a function $f_0$ ? It means $f_0(M)\subset U$ ?
