Medians of Discrete Sets according to a Linear Distance

In this paper, we present some results concerning the median points of a discrete set according to a distance defined by means of two directions p and q. We describe a local characterization of the median points and show how these points can be determined from the projections of the discrete set along directions p and q. We prove that the discrete sets having some connectivity properties have at most four median points according to a linear distance, and if there are four median points they form a parallelogram. Finally, we show that the 4-connected sets which are convex along the diagonal directions contain their median points along these directions. keywords: discrete set, linear distance, median point, projection, connected set, convexity.