We consider the domain of non-empty convex andcompact subsets of a finite dimensional Euclidean space torepresent partial or imprecise points in Computational Geom-etry. The convex hull map on such imprecise points is givendomain-theoretically by an inner and an outer convex hull. Weprovide a practical algorithm to compute the inner convex hullwhen there are a finite number of convex polytopes as partialpoints. A notion of pre-inner support function is introduced,whose convex hull gives the support function of the innerconvex hull in a general setting. We then show that the convexhull map is Scott continuous and can be extended to finitelygenerable subsets, represented by the Plotkin power domain ofthe underlying domain. This in particular allows us to computethe convex hull of attractors of iterated function systems infractal geometry. Finally, we derive a program logic for theconvex hull map in the sense of the weakest pre-condition fora given post-condition.