alexandrov geometry - Example of non-closed convex hull in a CAT(0) space

There are such examples already in Riemannian world!
In fact in any generic Riemannian manifold of dimension $ge3$ convex hull of 3 points in general position is not closed.
BUT it is hard to make explicit and generic at the same time :)

If it is closed then there are a lot of geodesics lying in its boundary --- that is rare!
To see it do the following exercise first: Show that in generic 3-dimensional manifold, arbitrary smooth convex surface contains no geodesic. (Here geodesic = geodesic in ambient space.)

To make word "generic" more clear: show that any metric admits $C^infty$-perturbation such that above property holds.

Semisolution:
Assume that a geodesic $gamma$ lies in the boundary of a convex set $K$ with smooth boundary. Let $N(t)$ be the outer normal vector to $K$ at $gamma(t)$. Note that $N(t)$ is parallel.
Further note that from convexity of $K$ we get that for any Jacoby field $J(t)$ such that

$$langle N(t_0),J(t_0)ranglele 0 text{and} langle N(t_1),J(t_1)ranglele 0,$$
we have
$$langle N(t),J(t)ranglele 0 text{if} t_0<t<t_1.$$
Note that this condition does not hold if the curvature tensor on $gamma$ is generic.

P.S. Roughly it means that convex hulls in Riemannian world are too complicated. But I know one example where it is used, see Kleiner's An isoperimetric comparison theorem.
But he is only using that Gauss curvature of non-extremal points on the boundary of convex hulls is zero...

Appendix. (A construction of convex hull.) To construct convex hull you can do the following: start with some set $K_0$ and construct a sequence of sets $K_n$ so that $K_{n+1}$ is a union of all geodesics with ends in $K_n$. The union $W$ of all $K_n$ is convex hull. Now assume it coincides with its closure $bar w$. In particular if $xinpartialbar W$ then $xin K_n$ for some $n$. I.e. there is a geodesic in $bar W$ passing through $x$ (if $xnotin K_0$). From convexity, it is clear that such geodesic lies in $partial bar W$...

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