We find conditions on the order on subsets of a finite set which are necessary and sufficient for the relative ranking of any two subsets in this order to be determined by their extreme elements relative to an abstract convex geometry. It turns out that this question is closely related to the rationalisability of path independent choice functions by hyper-relations.

Euclidean geometry on a computer is concerned with the translation of geometric concepts into a discrete world in order to cope with the requirements of representation of abstract geometry on a computer. The basic constructs of digital geometry are digital lines, digital line segments and digitally convex sets. The aim of this paper is to review some approaches for such digital objects. It is s...

The algorithms of computational geometry are designed for a machine model with exact real arithmetic. Substituting floating point arithmetic for the assumed real arithmetic may cause implementations to fail. Although this is well known, there is no comprehensive documentation of what can go wrong and why. In this extended abstract, we study a simple incremental algorithm for planar convex hulls...