On the Average Distances in a Circular Disc

Abstract

Using the $l^p $ notion of distance in the Cartesian plane, and assuming a uniform density of locations on l circular disc, we consider the resulting distance to any specified point of this domain, and we determine the first two moments of this random variable for $p = 1,2,\infty $. We find the maxima and minima of these average distances and their ratios, hence show their almost exact proportionality over the disc. Situations motivating these results include traffic flow on a rectangular street grid in a circular city and physical design of certain computer systems in two dimensions.