Consider a complete Riemannian manifold $M$. We know that it admits an exhaustion in compact subsets. My question is: can I assume all these subsets to be convex, by some operation of geodesic convex hull? Or, if you prefer: is the smallest convex subspace containing a compact subspace $C$ in a complete Riemannian manifold compact (bounded should suffice)?
Thank you in advance.
Start with an equilateral triangle $\Delta_1$ (viewed as a $1$-dim object) of side length $1$, take the three mid points of the sides of $\Delta_1$, attach to them the three vertices of an equilateral triangle $\Delta_2$ of side length $1/2$, take the three mid points of the sides of $\Delta_2$, attach to them the three vertices of an equilateral triangle $\Delta_3$ of side length $1/3$, ... In this way we get a non-compact graph $G$ so that if there is a convex exhaustion $K_1\subset K_2\subset ...$, then for sufficiently large $n$ it must be $K_n=G$.
Then try to make $G$ into a non-compact, complete $2$-manifold $M$ by changing each edge into very thin tubes and do tube connections near the branching points of $G$ ...