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I know the following equivalent definitions of the Convex Hull:

1) It is the smallest convex set containing the points

2) If the set is discrete, then every point in the sets convex hull is a convex combination of the points in the set.

3)It is the intersection of all the half-spaces containing the set.

because of the metric on the poincare disk I am trying to use (3) as my main tool to find the convex hull, and I have started reading on polyotopes.

My question is what is the convex hull of points that lie on the same geodesic in the poincare disk? My guess is just the geodesic they are on, but I'm not sure that this agrees with (3)

Any pointers?