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?

**UPDATE**: The answer to Question 1 is NO, as explained by Igor Rivin below.

**UPDATE**: The answer to Question 2 is YES, as claimed by John Harvey below. Originally it was proved by I.M. Liberman in "Geodesic lines on convex surfaces", C. R. (Doklady) Acad. Sci. URSS (N.S.) 32, (1941). 310–313.
The proof for 2-dimensional hypersurfaces is also reproduced in the book "Intrinsic geometry of convex surfaces" by A.D. Alexandrov, see Ch. IV, $\S$ 6, Theorem 1.