Let $M^n$ be the boundary of a convex compact set in $\mathbb{R}^{n+1}$ with non-empty interior.

**Question 1.** Is $M$ geodesically complete, i.e. is it true that every geodesic (= locally shortest path) can be extended infinitely in both directions? Will it still be true if the convex set is not necessarily compact, but only closed.

**Question 2.** Is it true that every shortest path on $M$ has both left and right first derivatives at every point?