TITLE: Minimizing geodesic on a convex surface
QUESTION [16 upvotes]: Let $\Sigma$ be a smooth convex surface in Euclidean 3-space
and $\gamma$ be a unit speed minimizing geodesic in $\Sigma$.
Assume that for some $a < b < c$, we have 
$$\gamma'(a)=\gamma'(b)=\gamma'(c).$$
Is it true that $\gamma'$ is constant on one of two intervals $[a,b]$ or $[b,c]$?
Comments

This is a simplification-variation of an other question I heard from Dima Burago.
It is not hard t construct an example of a minimizing geodesic such that $\gamma'(a)=\gamma'(b)$, but $\gamma'$ is not a constant on $[a,b]$. (See the example below.)
I would be also interested in the analog for $n$ points.
This paper: "Total curvature..." by Barany, Kuperberg, Zamfirescu is relevant.
Recently, in "On the total curvature..." by Nina Lebedeva and me we answered the original question of Dima Burago. The problem above remains open, likely the answer is "no".

Example. I will construct a convex polyhedron, but it is easy to smooth.
Consider polyhedron defined by 5 inequlaities: 
$$z\ge 0,\ \  |x+\alpha{\cdot}y|\le \alpha\  \ \text{and}\  \  z\pm\beta{\cdot}y\le \beta$$
and look at the minimizing geodesic between points $(0,1-\epsilon, \beta{\cdot}\epsilon)$ and $(0,-(1-\epsilon), \beta{\cdot}\epsilon)$. For appropriately chousen $\alpha$, $\beta$ and $\epsilon$ this minimizing geodesic will pass through the faces in this order $$\{z+\beta{\cdot}y= \beta\},\ \ \{x+\alpha{\cdot}y= \alpha\},\ \  \{z=0\},\ \ \{x+\alpha{\cdot}y= -\alpha\},\ \ \{z-\beta{\cdot}y= \beta\}$$ 
and it will have the same velocity vector $(1,0,0)$ on both faces $\{z+\beta{\cdot}y= \beta\}$ and  $\{z-\beta{\cdot}y= \beta\}$.

REPLY [7 votes]: Just to help visualize the example, if 
I have interpreted the description correctly, 
this is one view, for $\alpha=\beta=1$, and $\epsilon=\frac{1}{16}$.



In this view, the $x$-axis is horizontal, the $z$-axis vertical.
Addendum. The shortest path (yellow) from $p_1$ to $p_2$ follows the face sequence (Bk, R, Bt, L, F).


 I tried to indicate (in the nearly invisible short dashed lines) the initial direction of the path
on the (mauve) Bk and F faces, in their planar unfoldings.  I should note that these initial
velocities are not exactly $(1,0,0)$, which presumably is only achieved by "appropriately chosen"
$\alpha, \beta, \epsilon$.