On approximating the distribution of random distances within and between certain regions of space

A moment-based methodology is proposed for approximating the distribution of the distance between two random points belonging to sets that are composed of rectangles or rectangular parallelepipeds. The resulting density approximants are expressed in terms of Jacobi orthogonal polynomials. Two norms are being considered: the L1 norm referred to as the Manhattan distance and the L2 norm corresponding to the Euclidean distance. Several illustrative examples will be presented and certain applications to transportation and routing problems will be pointed out. The results can be readily extended to determine the distribution of the distance between a fixed point or a line segment and sets in ℝ or ℝ³, and to sets belonging to higher dimensions as well.