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. 

**Hint:** Use Jacobi fields to show the following:
If geodesic $\gamma$ lies in a convex surface then curvature tensor along $\gamma$ is *very* special.

**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...