The convex hull of finitely generable subsets and its predicate transformer

Abstract

We consider the domain of non-empty convex and compact subsets of a finite dimensional Euclidean space to represent partial or imprecise points in computational geometry. The convex hull map on such imprecise points is given domain-theoretically by an inner and an outer convex hull. We provide a practical algorithm to compute the inner convex hull when there are a finite number of convex polytopes as partial points. A notion of pre-inner support function is introduced, whose convex hull gives the support function of the inner convex hull in a general setting. We then show that the convex hull map is Scott continuous and can be extended to finitely generable subsets, represented by the Plotkin power domain of the underlying domain. This in particular allows us to compute, for the first time, the convex hull of attractors of iterated function systems in fractal geometry. Finally, we derive a program logic for the convex hull map in the sense of the weakest pre-condition for a given post-condition and show that the convex hull predicate transformer is computable.