Let $M$ be a smooth Riemannian manifold. Let $X\subset M$ be a closed path connected subset which has curvature bounded below in the sense of Alexandrov with respect to the induced intrinsic metric. Let $\gamma\colon [a,b]\to X$ be a shortest path in $X$ (parameterized by its length). Let $\iota\colon X\to M$ denote the natural imbedding.

**Question.** Is it true that $\iota\circ\gamma\colon [a,b]\to M$ has one sided derivatives *everywhere*?

**Remark.** The question has positive answer at *almost* every point of $[a,b]$ since $\iota\circ \gamma$ is a Lipschitz map, and such maps have first derivative almost everywhere.

**ADDED:** In the special case when $M=\mathbb{R}^n$ is the Euclidean space, and $X$ is a convex hypersurface, the answer is positive and due to I.M. Liberman (1941).